WASHINGTON UNIVERSITY IN ST. LOUIS

Department of Chemistry

Dissertation Examination Committee:Lee Sobotka, ChairRobert CharityWillem Dickhoff

Demetrios SarantitesJacob Schaefer

CONTINUUM NUCLEAR STRUCTURE OF LIGHT NUCLEI

by

Kyle Wayne Brown

A dissertation presented to theGraduate School of Arts and Sciences

of Washington University inpartial fulfillment of the

requirements for the degreeof Doctor of Philosophy

August 19, 2016

Saint Louis, Missouri

©2016, Kyle W Brown

Contents

List of Figures v

List of Tables xvii

Acknowledgments xix

Abstract xx

1 Introduction 1

1.1 Nuclear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Stellar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Exotic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Nuclear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.3 Proton Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.4 Decay Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Experimental Methods 18

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

2.2 Invariant-mass Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 The Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.2 Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Isobaric Analog State of 8B 32

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Decay mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 16Ne Ground State 41

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Excitation Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Ground-state Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Three-body Energy-Angular Correlations . . . . . . . . . . . . . . . . . . . . 47

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 16Ne Excited States 53

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 14O+p+p Excitation spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3 Width of the 2+ state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4.1 Simplified decay model . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4.2 Three-body calculations . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4.3 Three-body calculations . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.4 Simplified decay model estimates . . . . . . . . . . . . . . . . . . . . 62

ii

5.4.5 Outlook for 2+ state width . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Correlations for 2+ state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.5.1 Background contribution to correlations . . . . . . . . . . . . . . . . 66

5.5.2 Correlations in the simplified model . . . . . . . . . . . . . . . . . . . 69

5.5.3 Comparison with three-body calculations . . . . . . . . . . . . . . . . 71

5.6 Decay mechanism for 2+ state . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.7 Previous Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.8 Higher Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Isobaric Analog State in 16F 84

6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.2 Two-proton decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.3 Isospin non-conserving decays . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4 Gamma decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7 A = 7 IMME 97

7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.3 R-matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.4 Resonance energy definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.5 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.6.1 General Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.6.2 7LiIAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.6.3 7BeIAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.6.4 7Bg.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

iii

7.7 Isobaric Multiplet Mass Equation . . . . . . . . . . . . . . . . . . . . . . . . 118

7.8 Variation of Spectroscopic Strength . . . . . . . . . . . . . . . . . . . . . . . 122

7.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8 Assorted States 127

8.1 9C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.2 8B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.3 17Na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.4 17Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

9 Summary 141

Bibliography 145

iv

List of Figures

1.1 Chart of nuclides. Proton number increases towards the top of the figure, and

neutron number increases towards the right. Stable nuclei are shown as black

squares, and the radioactive nuclei are shown as colored squares, indicating

their dominate mode of decay. Neutron and proton closed shells are indicated

by the blue rectangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Big Bang Nucleosynthesis reaction network. All arrows are labeled by the re-

action they represent. Initially only neutrons and protons exist, but deuterium

(D) is made through the neutron capture reaction, p(n,γ)D. From there other

reactions occur to produce: tritium (T), 3,4He, 7Be, 7Li. . . . . . . . . . . . 5

1.3 Carbon-Nitrogen-Oxygen cycle. The 12C acts as a catalyst to the same net

reaction as in the pp-chain. While this process is occurring in stars, 5 carbon,

nitrogen and oxygen isotopes are present and available for other nucleosyn-

thetic processes (i.e. alpha capture). . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Residuals (experimental binding energy - LDM prediction) as a function of

neutron number (N). Different curves correspond to different elements (Z).

Large deviations from the data are seen at N =28, 50, 82, and 126. To keep

these deviations in perspective, the total binding energies are roughly 8 MeV

× the number of nucleons. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

v

1.5 Single-particle levels for the IPM with (from left to right) (a) the harmonic

oscillator, (b) infinite square well, (c) finite square well and (d) Wood-Saxon

potentials. The final column is for a Wood-Saxon potential with the spin-orbit

term included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Isobar diagram for A=7. Levels are labeled by their energy, spin, and isospin

when known, with intrinsic widths indicated by the size of the cross-hatching.

States with identical quantum numbers are connected by dashed lines to their

analogs. Thresholds for particle decays are labeled with their energy relative

to the ground state of the given nucleus. Figure taken from Ref [1]. . . . . . 13

1.7 Partial levels schemes for the different classifications of 2p decay. Direct 2p

decay is shown in (a) and (c), sequential in (b) and democratic in (d,e). Figure

taken from Ref [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.8 Jacobi vectors for three particles in coordinate and momentum spaces in the

“T” and “Y” Jacobi system. The X and Y vectors define each system. Figure

taken from Ref [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1 Schematic of the experimental setup at Michigan State University. Stable

beams (green) are accelerated first by the K500 and then the K1200 cyclotrons.

The stable beam impinges on a 9Be target, where it is fragmented. The

fragments are separated by the A1900 magnetic fragment separator based on

their mass-to-charge ratio. These radioactive beams (red) are transported

to the experimental area where they undergo further reaction to create the

unbound products. Charged particles are detected with HiRA, and γ rays are

detected by CAESAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Energy deposited in a silicon (∆E) vs energy deposited in a CsI(Tl) (E)

for a typical outer HiRA telescope. The silicon energy has been calibrated

and the CsI(Tl) energy is uncalibrated. Different bands are labeled by their

corresponding isotope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

vi

2.3 HiRA Array viewed from upstream. . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Layout of CAESAR detectors. (left) cross-sectional view perpendicular to

beam axis showing rings B and F and (right) cross-sectional view parallel to

beam axis showing all ten rings and target position (dot with vertical line

through it). Taken from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Calibrated alpha energy spectrum for a typical silicon strip in the front (hole

collecting) side. The FWHM of the 8.785 MeV peak is 72 keV. . . . . . . . . 28

2.6 Pulser calibration spectrum for a typical silicon strip on the front side. The

pulses set in uniform charge increments and a non-linearity of the electronics

above half scale is clearly seen. The pulse corresponding to the peak near

channel number 10,000 was run for three times as long as the other pulses. . 30

3.1 Set of levels relevant for the decay of the IAS in 8B. The levels are labeled

by their spin-parity (Jπ) and isospin (T) quantum numbers. Colors indicate

isospin allowed transitions. The width of the Jπ = 7/2− state in 7Be is not

known, but assumed to be wide as in the mirror. . . . . . . . . . . . . . . . 34

3.2 The excitation-energy spectrum from charged-particle reconstruction deter-

mined for 8B from detected a) 2p-6Li and b) 2p-α-d events.These energies

assume no missing decay energy due to γ-ray emission. . . . . . . . . . . . . 35

3.3 Gamma-ray energies measured in coincidence with p-p-6Li events. This spec-

trum includes add-back from nearest neighbors and has been Doppler-corrected

eventwise. The peak at about 3 MeV is when one of the two 511 keV γ rays

from the annihilation of the pair produced positron escapes detection in CAE-

SAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vii

3.4 Left panel (a-d) are the projected three-body correlations from the decay of

8BIAS to the 2p+6LiIAS exit channel in the (a)(c) T and (b)(d) Y Jacobi

systems. Energy correlations are shown in (a) and (b), angular correlations

in (c) and (d). The right panel shows the same as the left, but now the

correlations are associated with the first step of 8Cg.s decay to 2p+6Be. . . . 39

4.1 Experimental spectrum of 16Ne decay energy ET reconstructed from detected

14O+p+p events. The dashed histogram indicates the contamination from

15O+p+p events. The smooth curves are predictions (without detector reso-

lution) for the indicated 16Ne states. The inset compares the contamination-

subtracted data to the simulation of the g.s. peak for Γ = 0, ftar = 0.95, where

the dotted line is the fitted background. . . . . . . . . . . . . . . . . . . . . 45

4.2 Comparison of simulated line shapes of the best fit (red curve) and 3σ upper

limit (blue dashed curve) to the data. See text for description of fit parameters. 46

4.3 Energy-angular correlations for 16Neg.s.. Experimental and predicted (MC

simulations) correlations for Jacobi “T” and “Y” systems are compared. . . . 49

4.4 The convergence of the predicted (a) decay width and (b) energy distribution

in the “T” system on Kmax (maximum principal quantum number of hyper-

spherical harmonics method). The asymptotic decay width of 16Ne assuming

exponential Kmax convergence is given in (a) by the dashed line. . . . . . . . 50

4.5 Panels (a)-(d) show energy distributions in the Jacobi “Y” system where (a)

gives the sensitivity of the predictions to ρcut, (c) to the 15Fg.s. properties,

and (d) to the decay energy ET . Panels (e), (f) show energy distributions in

the Jacobi “T” system where (e) gives the sensitivity to ρcut. The theoretical

predictions, after the detector bias is included via the MC simulations, are

compared to the experimental data in (b) and (f) for the “Y” and “T” systems

respectively. The normalization of the theoretical curves is arbitrary, while

the MC results are normalized to the integral of the data. . . . . . . . . . . . 51

viii

4.6 The core-proton relative-energy distribution (“Y” system) obtained by classi-

cal extrapolation started from different ρmax values. . . . . . . . . . . . . . . 52

5.1 Low-lying levels of 16Ne observed in this work and Ref. [5] and levels important

for proton and multi-proton decay to 15F, 14O, and 13N states. The exper-

imental uncertainty of the 15F g.s. energy is indicated by hatching. Note a

reduction 1.5 in the scale above the 13N+p+p+p threshold. All level energies

are given with respect to the 14O+p+p threshold. . . . . . . . . . . . . . . . 56

5.2 Distribution of 16Ne decay energy determined from detected 14O+p+p events

showing peaks associated with the ground and first excited states of 16Ne.

The solid curve shows a fit to the first excited state where the dashed curves

indicates the fitted background under this peak. Two background gates, B1

and B2, are shown which are used to investigate the background contributions

to the measured decay correlations. . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Convergence of the width calculations for the 16Ne 2+ state. (a) Kmax con-

vergence at a fixed KFR. (b) KFR convergence at a fixed Kmax. Exponential

extrapolations (separately done for odd and even KFR/2 values) point to a

width of around Γ = 56 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Comparison of the experimental Jacobi (a) “Y” and (d) “T” correlation dis-

tributions to the those of the three-body model [(b) and (e)] and those from a

sequential decay simulation [(c) and (f)]. The effects of the detector efficiency

and resolution in the theoretical distributions have been included via Monte

Carlo simulations. (g) Shows the relative orientation and magnitude of the

two proton velocity vectors for the peak regions indicated by blue circles in

panel (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

ix

5.5 Comparison of projected Jacobi (a) “Y” and (c) “T” energy distributions for

peak-gated events and the events in the neighboring background gates B1

and B2. In (b) and (d) the experimental projections have been background

subtracted (see text) and are compared to the predictions of the three-body

model before (dashed curves) and after (solid curves) the effects of the detector

bias and acceptance was included. The arrows in (a) show the predicted

locations of the maxima from Eq.(5.1) for the mean energies of each of the

three gates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.6 Experimental energy distribution for 16Ne 2+ state in the “Y” Jacobi system

(data points) is compared with distributions calculated by the R-matrix-type

expression of Eq. (5.1) for the different s1/2 g.s. resonance energies Er in 15F

listed in the figure. The detector efficiency and resolution were included via

the Monte Carlo simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.7 Correlation density W (ρ, θρ) for 16Ne 2+ state WF Ψ(+) in the “Y” Jacobi

system. Panels (a) and (b) show the evolution of the predicted hyperangular

distribution with hyperradius on two different scales. Yellow curves in (a)

provide the levels with constant absolute values of X and Y vectors indicated

by the attached numbers (in fm). Blue (dashed) and pink (solid) arrows

qualitatively indicate the classical paths connected with sequential and the

“tethered” decay mechanisms correspondingly. The blue frame in the panel

(b) shows the scale of panel (a). Panel (c) illustrates the arrangement of the

X and Y vectors for the “Y” Jacobi system, see Eqs. (5.3) and (5.4). . . . . 73

5.8 Schematic of the tethered decay mechanism. Trajectory of two protons corre-

sponding to solid (pink) path in Fig. 5.7. The configurations labeled by the

Roman numerals correspond to the same configurations denoted in Fig. 5.7.

The p-p center-of-mass motion path rx = ry is collinear with Y vector for the

Jacobi “T” coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . 79

x

5.9 Distribution of 16Ne decay energy determined from detected 13N+p+p+p

events. Gates are shown on the two observed peaks which are used in the

analysis shown in Figure 5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.10 Distribution of 14O decay energy determined for all possible 13N+p subsets of

each detected 13N+p+p+p event. Pannel (a) is for the gate on the E ′T = 5.21

MeV state in 16Ne while (b) is for the E ′T = 7.60 MeV state. The vertical

dashed lines show the location of the first five excited states in 14O. . . . . . 82

6.1 Deviation from the fitted quadratic form of the IMME for the lowest T = 2

states in the A = 16 isobar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Partial level scheme for the decay of 16FIAS (level in green). Colored levels

indicate isospin allowed decays from the 16FIAS. All energies are relative to

the ground state of 16F. The inset shows an expanded level scheme for the

low-lying states of 16F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 (a) Excitation energy spectrum of 16F from all detected 2p+14N events. The

inset is shows the same histogram expanded in the region around the expected

16FIAS peak. (b) γ ray energy spectrum measured in coincidence with events

inside of the gate indicated in (a) with the blue dashed lines. The excepted

photopeak position (2.313 MeV) is marked with the red dashed line. . . . . . 89

6.4 Excitation energy spectra for all detected (a) α+12N events and (b) 3He+13N

events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.5 Excitation energy spectra for all detected p+15O events. The blue arrow

indicated the energy of the T = 1 state from the blue path, see text, if the

γ-ray energy were added. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

xi

6.6 (a) γ energy spectrum in coincidence with all p+15O events. (b) 16F excitation

energy spectrum of p+15O events. This is the same spectrum shown in Fig.

6.5 but with a linear ordinate. The blue dashed line marks the position of the

first-excited state in 16F. The red dashed histogram is the excitation energy

spectrum gated on the gamma rays in gate G1. . . . . . . . . . . . . . . . . 93

6.7 Partial level scheme of 16F . The red arrows indicate a possible isospin-

conserving decay mode for the decay of 16FIAS, and the blue mode indicates

the possible decay mode of a T = 1 state of 16F. . . . . . . . . . . . . . . . 94

7.1 (Color Online) Elastic scattering phase shifts for p+6He (δp) and n+6LiIAS

(δn) scattering extracted for the 7LiIAS resonance. The energy is relative to

the proton threshold Ep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 (Color Online) Set of levels relevant for the decay of the T = 3/2 members of

the A = 7 quartet, with (a)-(d) arranged with increasing TZ . The levels are

labeled by their spin-parity (Jπ) and isospin (T) quantum numbers. Observed,

isospin-allowed decays are shown with solid red lines, and unobserved, isospin-

allowed decays are shown with dashed red lines. Levels to which particle decay

is allowed are shown in red. The properties of the A=6 states are listed in

Table 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.3 (Color Online) Overlap function predicted by the GFMC and VMC method

for the 7Heg.s. → n+6He channels. For the the VMC methods, results are

shown for decays to the ground (Jπ=0+) and first excited state (Jπ=2+)

state of 6He, while only the ground state is shown for the GFMC method.

The Jπ=0+ channel is unbound and the dashed curves show single-particle

overlaps with the correct asymptotic behavior that match with the theoretical

calculations in the interior region (see text for details). . . . . . . . . . . . . 109

xii

7.4 (Color Online) The excitation energy spectrum for 7Li reconstructed from

all p+6He events. The red solid curve is our R-matrix fit with S2+=0 (θ2c

taken from GFMC [6]). Two almost identical fits were made with different

background contributions. The dotted and dashed curves indicated the fitted

background obtained using parametrizations. b1 and b2, respectively. . . . . . 113

7.5 (Color Online) The Doppler-corrected γ-ray spectrum measured in coincidence

with detected p+6Li events. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.6 (Color Online) (a) Reconstructed minimum excitation-energy spectrum (miss-

ing the γ energy) for the p+6Li events in coincidence with a 3.563 MeV γ ray

(G1 in Fig. 7.5). The blue dashed histogram indicates a contamination from

8B+p+p+γ events where one proton misses the array. The red dot-dashed

histogram is a background from the γ-ray gate (B1 in Fig. 7.5). (b) The

points are the background-subtracted excitation-energy spectrum. The red

solid curve is a simulation of the decay from 7BeIAS using a R-matrix line

shape (θ2c taken from GFMC [6]) with the experimental resolution included. . 115

7.7 (Color Online) Invariant mass of 7B from all detected 3p+α events. Two R-

matrix fits with S2+=0 are shown as the red solid curve using both background

parameterizations. These two fits are almost identical and cannot be differen-

tiated, but their fitted backgrounds are somewhat different. The blue dotted

and blue dashed curves were obtained with the b3 and b4 parameterizations,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.8 (Color Online) R-matrix line shape for 7B with (without) the contribution of

the Jπ = 2+ state included shown as the red dotted (blue solid) curve.The

location of the resonance energy input to the R-matrix (E(1)r ) is shown in

green, and the resonance energy from the pole of the S-matrix (E(3)r ) is shown

in magenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xiii

7.9 (Color Online) Deviation from the fitted quadratic form of the IMME for the

lowest T = 3/2 states in the A = 7 isobar. The data points for the E(1)r and

E(3)r definitions are shifted by 0.1 and 0.2, respectively, along the abscissa for

clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.10 (Color Online) Ground-state spectroscopic factor as a function of isospin pro-

jection (TZ) using θ2c from (a) Green’s Function Monte Carlo and (b) VMC.

The open blue data points have S2+ = 0 and the solid magenta data points

have S2+ = 0.75. The top- and bottom-pointing triangles are from [7] and [8],

respectively. The horizontal lines show the predicted spectroscopic factors for

7Heg.s. from the respective models. . . . . . . . . . . . . . . . . . . . . . . . . 123

7.11 (Color Online) Comparison of the width of the T = 3/2 resonance for A = 7

as a function of TZ from the FWHM of the R-matrix line shape and extracted

from the S-matrix. The contribution of including the Jπ = 2+ is also shown.

The dotted line is a quadratic fit. . . . . . . . . . . . . . . . . . . . . . . . . 125

8.1 Partial level scheme for the proton decay of 9C and 8B. Levels are labeled by

their spin and parity (Jπ) when known. New levels, widths, or decay modes

are plotted in magenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.2 Invariant-mass spectrum of 9C from all detected p+8B events.The blue dashed

and green dotted lines are R-matrix simulations for the 1/2− and 5/2− states,

each curve has been scaled down by a factor of two for clarity. The orange

dot-dashed line is one parameteriza0.0tion of the background. The sum of the

background and two simulations is plotted as the red solid line. . . . . . . . 128

8.3 (a) Excitation energy spectrum for 2p+7Be events plotted as a function of the

excitation energy for the intermediate (8B) for each p+7Be combination. (b)

Excitation energy spectrum for 9C from all detected 2p+7Be events (black).

The red and blue spectra are the excitation energy spectrum for 9C gated on

gates G1 and G2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 129

xiv

8.4 Excitation energy spectrum of 8B from all p+7Be events (black solid line).

The peaks corresponding to the decay of the 1+ state are fit with the sum of

two Gaussian fits and a linear background (dashed line). . . . . . . . . . . . 131

8.5 The black solid histogram is the γ-ray energy spectrum gated on the small

peak in the 8B invariant-mass spectrum. The blue dashed histogram is a

CAESAR γ-ray background. . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.6 Decay energy spectrum of 17Na from all detected 3p+14O events. The blue

arrow marks the average energy of the peak, and the red arrow marks the

ground-state energy predicted from mass systematics. . . . . . . . . . . . . 134

8.7 Partial level scheme for the 2p decay of 17Ne. States are labeled by their spin,

parity, and energy relative to the ground state of 17Ne. . . . . . . . . . . . . 135

8.8 Excitation energy spectrum of 17Ne from all detected 2p+15O events. The

solid red curve is an R-matrix fit to the 1.7 MeV (Jπ = 5/2−) second-excited

state of 17Ne with a quadratic background (red dashed line). Peaks corre-

sponding to the Jπ = 5/2+ and 9/2− states are seen at higher energy. . . . . 136

8.9 (a) Jacobi Y energy variable for the 2p decay of 17Ne second-excited state. Red

solid (blue dashed) curves are Monte Carlo simulations for the decay through

the ground state (first-excited state) of the 16F intermediate. (b) Jacobi Y

energy-angular correlations for 17Ne second-excited state. (c) and (d) are

Monte Carlo predictions for the decay through the ground and first-excited

state of 16F respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.10 R-matrix simulations for the 2p decay of 17Ne second-excited state through

16F (a) ground-state and (b) first-excited state. The line shape for the second

proton p2 (black), the barrier penetration factor for the first proton p1, and

their product (blue) are displayed for both decay paths. . . . . . . . . . . . . 140

xv

9.1 Chart of nuclides. Proton number increases towards the top of the figure, and

neutron number increases towards the right. Stable nuclei are shown as black

squares, and the radioactive nuclei are shown as colored squares, indicating

their dominate mode of decay. Nuclei studied in this work are labeled. . . . . 142

xvi

List of Tables

2.1 Alpha energies, half-lives, and branching ratios for the 228Th decay chain. α′s

with less than 5% intensity are not included in this table. α0 and α1 denote

decay to the ground state and first-excited state of the daughter, respectively. 29

2.2 γ-ray energies (Eγ) and gamma fraction per decay (Iγ) for the sources used to

calibrate CAESAR. γ rays from the Am-Be source come from the 9Be(α,γ)12C

reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Simplified-decay-model estimates of important configurations of the 16Ne 2+

WF for D3=1.0. The total decay width according to Eq. (5.2) is provided in

the line “Total”. The sensitivity of the three-body width to variations of the

15F g.s. energy is illustrated by the three bottom lines of the Table. . . . . . 64

5.2 The properties of 0+ g.s. and first 2+ states of 16Ne obtained in the recent

neutron knockout reaction studies with 17Ne beam. Energies and widths are

in MeV. All the ET values are provided relative to the 14O+p+p threshold. . 78

6.1 Decay energies (ET ) and barrier penetration factors (Pℓ) for the decay channels

relevant to this work. The barrier penetration factors for 2p decays are calcu-

lated assuming that each proton takes away 1/2 ET . Isospin non-conserving

decays are listed in the lower section of the table. . . . . . . . . . . . . . . . 88

7.1 Excitation energy and widths of the A=6 daughter states from Ref. [9]. . . . 108

xvii

7.2 Resonance energies and decay widths extracted from the R-matrix analyses

of the A = 7 cases. The E(2)r resonance energy is channel dependent and thus

values are listed from both proton (p) and neutron (n) exit channels. . . . . 110

7.3 Spectroscopic factors S0+ extracted from the R-matrix fits in this work. Re-

sults are listed for both the θ2s.p. values obtained from the GFMC and VMC

overlap functions and for S2+=0 and S2+=2.0 . . . . . . . . . . . . . . . . . 111

7.4 Comparison between the measured excitation energies (E∗exp) and the litera-

ture values (E∗ENSDF ) [9] for narrow resonances used to estimate the system-

atic uncertainties in the two experiments. The deviation is given as ∆E∗ =∣∣E∗exp − E∗

ENSDF

∣∣. For the 7Bg.s. experiment, we list equivalent results for the

total decay energy ET of 6Be. . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.5 Mass excesses for the A = 7, isospin T = 3/2 quartet using the E(1)r resonances

energies and the coefficients from the quadratic fit. . . . . . . . . . . . . . . 121

7.6 Comparison of the coefficients obtained from the cubic fits to the quartet

masses for the two definitions of resonance energy (E(1)r , and E

(3)r ) and from

the GFMC calculations of Ref. [10]. . . . . . . . . . . . . . . . . . . . . . . 121

9.1 New spectroscopic information for nuclear levels discussed in this work. Exci-

tation energies, intrinsic widths and decay modes are provided when measured.

For nuclear levels with multiple decay modes, branching ratios are given as

intensities out of 100 relative to the dominant decay mode. For ground-state

decays, decay energy is given in place of excitation energy. . . . . . . . . . . 144

xviii

Acknowledgments

First and foremost I would like to thank my outstanding advisor, Lee Sobotka, who made

my time in graduate school both memorable and enjoyable. His guidance has enabled me

to become the scientist I am today. I would also like to thank Bob Charity for being an

excellent mentor. I am grateful to him for the use of his analysis and simulation codes which

have been utilized heavily for this work. I would like to thank my committee for all of their

hard work.

I am eternally grateful to my parents, Karla and Gary Brown, whose constant love and

support has been the rock I needed to succeed.

I would like to thank the HiRA collaboration for their assistance during the two invariant-

mass experiments in this work. In particular I would like to thank Zbigniew Chajecki,

without whom the experiments would not have been possible. The assistance of Alexandra

Gade and Dirk Wiesshaar on use of the CAESAR array was greatly appreciated. I would

also like to thank the A1900 group at the NSCL for the production of the 9C and 17Ne

secondary beams.

The detailed understanding of the 2p decay of 16Ne would not have been possible without

the calculations of Leonid Grigorenko. His insight and discussions were an integral part of

this work. I would also like to thank Robert Wiringa for his calculations on the A = 7 isobar.

I would also like to thank my undergraduate advisor, Romualdo deSouza, and my mentor,

Sylvie Hudan. They were essential in providing the training needed to succeed early on in

my graduate studies.

xix

Abstract

In this work, the continuum decay of proton-rich nuclei with mass number A ≤ 17 was stud-

ied. The nuclear structure of these nuclei was assessed by measuring the decay energy, width,

and momentum correlations between the decay fragments. In particular the Isobaric Analog

State in 8B, the ground-state and excited states of 16Ne, the T = 3/2 states in the A = 7

isobaric chain, and the excited states of 9C and 17Ne were all studied. In addition a first

measurement of 17Na was made. These states decay by emission of 1 or more protons and

were populated through knockout reactions or inelastic scattering of 9C and 17Ne secondary

beams produced at the National Superconducting Cyclotron Laboratory. The charged parti-

cles were detected in the High Resolution Array (HiRA) which, in these studies, consisted of

14 Si-CsI(Tl) telescopes. Gamma rays measured in coincidence with charged particles were

measured in the 4π array of CsI(Na) called CAESAR.

xx

Chapter 1

Introduction

1.1 Nuclear Stability

The chart of nuclides, Fig. 1.1, is a plot of all known nuclei. Nuclei are displayed according

to their number of protons (p) and neutrons (n), with proton number (Z) plotted along the

vertical axis and neutron number (N) plotted along the horizontal axis. The stable nuclei,

shown as black squares in Fig. 1.1, have roughly equal numbers of protons and neutrons for

low mass number (A = N+Z) nuclei, and the highest mass stable nuclei have roughly 40 %

protons. This is due to the competition between the increasing Coulomb interaction between

protons, that drives stability towards neutron-rich nuclei and the nuclear asymmetry energy

Easy ∝ C(N − Z)2

A(1.1)

that drives stability towards N = Z. Off the line of stability are radioactive nuclei that

can be accessed via nuclear reactions. In general, as one moves away from stability along

isotopes (same Z) or isotones (same N) the beta-decay half-lives become shorter (blue and

pink squares of Fig. 1.1). The binding energy of the last neutron or proton also decreases as

one moves towards neutron-rich or neutron-deficient nuclei, respectively. At some point the

binding energy of the last proton or neutron goes to zero, this is referred to as the drip-line.

1

Nuclei beyond the drip lines exist only as short-lived resonances, and will decay by emission

of one or more nucleons (orange and purple squares). At the high mass end of the chart of

nuclides are nuclei which decay either by alpha emission or spontaneous fission (yellow and

green squares), which are not the subject of this work.

Nuclei with half-lives much shorter than the age of the earth that are not still being

produced through some physical process today (i.e. alpha decay chains or atmospheric

production), must be created in the laboratory. Knockout and transfer reactions allow

us to study nuclei one or a few nucleons from stability. Nuclei further from stability can

be produced by fusion or fragmentation reactions or as the products of fission. Due to

the decreasing slope of the line of stability, fusion and fission allow us to study neutron-

deficient and neutron-rich species, respectively. Fragmentation reactions allow for a much

wider range in asymmetry (n/p ratio), but can only be used to study nuclei with mass less

than the nucleus being fragmented. Nuclei very far from stability, often called exotic nuclei,

are very difficult to produce. The most exotic are those nuclei near or beyond the proton and

neutron drip lines. These unstable nuclei can be very short lived (nanoseconds or shorter).

Measuring the properties of these nuclei (half-lives, decay modes) plays an important role in

our understanding of element production in the universe as nucleosynthesis often progresses

through these unstable configurations of neutrons and protons. To produce these nuclei

in the laboratory one can use a two-step process. The first step could be fragmentation

to produce nuclei which are a few nucleons from stability, with half-lives of milliseconds

(particle bound but unstable to β decay), and then a second step (executed proximally to

appropriately designed experimental equipment) to produce the very short-lived (particle-

unbound) nucleus. This will be the method employed in the present work.

2

Figure 1.1: Chart of nuclides. Proton number increases towards the top of the figure, andneutron number increases towards the right. Stable nuclei are shown as black squares, andthe radioactive nuclei are shown as colored squares, indicating their dominate mode of decay.Neutron and proton closed shells are indicated by the blue rectangles.

3

1.2 Nucleosynthesis

1.2.1 Big Bang

In the very earliest times in the universe, it was too hot for even protons or neutrons to exist.

As the universe expanded and cooled, neutrons and protons formed and were in equilibrium

through the following reactions,

νe + p e+ + n (1.2)

νe + n e− + p. (1.3)

When the temperature drops to T = 109 K, the neutrinos decouple, and the neutrons and

protons fall out of equilibrium. While neutrons will beta decay to protons, the half-life is

long compared to the time-scale of Big Bang Nucleosynthesis (BBN). This allows neutron

reactions to take place during BBN. Figure 1.2 shows the reaction network that takes place

during BBN. The net result of this, the first act of nucleosynthesis, is about 75 % protons,

23 % 4He, by mass, and trace amounts of other very light nuclei. The universe continues

to expand and cool. When the temperature drops below that corresponding to the electron-

proton binding energy (13.6 eV, or at t = 377000 years), electrons bind to make atoms and

photons decouple and free stream. For nucleosynthesis the story pauses until the formation

of stars.

1.2.2 Stellar

Nucleosynthesis in the modern universe can be broken into two classes, stellar and explo-

sive. For all “main sequence” stars, like the sun, the dominant process is called the pp-chain

(proton-proton chain). This process converts four protons into an alpha particle (4He nu-

cleus) through a series of reactions, and in the process releases 27 MeV of energy (The

4

Figure 1.2: Big Bang Nucleosynthesis reaction network. All arrows are labeled by thereaction they represent. Initially only neutrons and protons exist, but deuterium (D) ismade through the neutron capture reaction, p(n,γ)D. From there other reactions occur toproduce: tritium (T), 3,4He, 7Be, 7Li.

5

products have that much less mass/energy than the reactants). The net reaction is as fol-

lows:

4p →→→ 4He++ + 2e+ + 2ν + 26.73 MeV. (1.4)

In stars heavier than the sun, this reaction proceeds through a process known as the CNO

cycle. This is a catalytic process that uses 12C “seed” nuclei, Fig. 1.3. This process produces

intermediate nuclei such as 15O, which in this cycle beta decays to 15N. Before this happens,

it can undergo a CNO breakout alpha capture reaction, 15O(α,γ)19Ne, which provides the

seeds for the rp-process.

When a star begins to run out of hydrogen, the core of the star collapses and the temper-

ature increases. This rise in temperature leads to the helium burning phase of a star’s life,

where three alphas come together to make 12C. This is a two step process in which two alphas

fuse to create 8Be, which is unbound and will decay back to two alphas rapidly. However if

another alpha particle encounters the 8Be before it can decay, they can fuse to create 12C in

a special Jπ = 0+ excited state called the Holye state. Four times in 10,000 this state will

decay by emitting two γ rays, de-exciting to the stable 12C ground state. The rest of the

time it will decay back to 8Be and then back to 3α’s. For lighter stars, this is the last stage

of their productive lives. When the helium is exhausted, they blow off their outer layers to

become white dwarfs. For more massive stars, the 12C and the material produced from the

CNO cycle can undergo a series of alpha capture reactions EAZ(α,γ)E

A+4Z+2 to produce nuclei

on or near the line of stability. These reactions are all exothermic until the Fe-Ni region, at

which point this process shuts off.

Moving beyond iron nuclei (in normal stellar life) requires a process known as the s-

process, or the slow neutron-capture process. This process takes the seeds created from the

alpha captures, and produces heavier nuclei through neutron capture, EAZ(n,γ)E

A+1Z . The

s-process is slow on the time scale of beta decay, so if these new nuclei are beta-unstable,

they will decay before another neutron-capture reaction can take place. Therefore only nuclei

6

Figure 1.3: Carbon-Nitrogen-Oxygen cycle. The 12C acts as a catalyst to the same netreaction as in the pp-chain. While this process is occurring in stars, 5 carbon, nitrogenand oxygen isotopes are present and available for other nucleosynthetic processes (i.e. alphacapture).

7

right along the line of stability can be created by this method. The neutron-capture reactions

require the presence of free neutrons, which have long since decayed from the time of BBN.

Active stars can produce free neutrons from

13C(α, n)16O and 22Ne(α, n)25Mg. (1.5)

This is the final process which can occur in stars during their normal quasi-stationary life.

For further production of heavy nuclei, and nuclei which the s-process cannot produce, more

explosive environments are required.

1.2.3 Exotic Processes

There are two main pathways for explosive nucleosynthesis, namely the r-process and the

rp-process. The r-process, or rapid-neutron capture process, has inverted kinetics to that of

the s-process. Here the neutron captures occur rapidly enough that the beta-unstable nuclei

created by this process cannot decay before they themselves undergo neutron capture. For

a large neutron-flux, nuclei almost out to the drip line will be produced. This process is

retarded when a closed shell is hit as the capture cross-section drops dramatically. While

the site of the r-process is not known experimentally, it is thought to be an explosive event

like a supernova or neutron-star merger. In both of these astrophysical events, the neutron

flux is high enough for this process to occur. When the event is over and the neutron flux

dies down, the nuclei created by this process will beta-decay back along an isobar, constant

A = N + Z, to the line of stability. The r-process is responsible for the creation of half of

the nuclei heavier than iron.

The rp-process can be thought of as the mirror of the r-process. Here a series of rapid

proton capture reactions create nuclei out to the proton drip line. Due to the Coulomb barrier

of the proton capture reactions, this process must occur in a high-temperature environment.

8

It is thought to occur in compact, binary systems like a neutron star paired with a main

sequence star. There, the intense gravity of the neutron star can pull matter off of the

companion star. This accretion of matter, largely protons and helium, builds a layer on the

surface of the neutron star that increases in temperature as the density increases. Eventually

enough material collects on the surface that a thermonuclear explosion can occur, triggering

temperatures high enough for the rp-process.

While the overall mechanisms for the r- and rp-processes are thought to be understood,

the majority of nuclei involved in the processes have never been studied experimentally. By

studying these exotic nuclei in the lab, we can better understand the reaction rates in these

astrophysical processes.

1.3 Nuclear Structure

1.3.1 Nuclear Models

Figure 1.4: Residuals (experimental binding energy - LDM prediction) as a function ofneutron number (N). Different curves correspond to different elements (Z). Large deviationsfrom the data are seen at N =28, 50, 82, and 126. To keep these deviations in perspective,the total binding energies are roughly 8 MeV × the number of nucleons.

9

In the 1930s, Carl Friedrich von Weizsacker introduced one of the first successful models

of the nucleus, called the Liquid-Drop Model (LDM). The LDM model treats the nucleus

as a semi-classical fluid of neutrons and protons, ie this model does not try to solve the

quantum mechanical problem. This can be considered a “macroscopic” model of the nucleus

because it starts with a term describing the bulk properties of infinite nuclear matter, and

then applies corrections due to the finite size of the nucleus. Corrections are needed for the

surface (nucleons on the surface do not “feel” the attractive force from all sides), the Coulomb

repulsion (every protons repels every other proton), and a correction for a neutron/proton

asymmetry . The asymmetry correction is the only quantum mechanical ingredient to the

LDM, and accounts for the fact that neutrons and protons should fill single-particle orbitals

in the same way that electrons do in atoms. Neutrons and protons fill these orbits separately,

and the orbits should be roughly the same for each particle type, so there is an energy penalty

for having different numbers of neutrons and protons. There is also a term which accounts for

the pairing of like nucleons. The binding energy predicted by the LDM can be summarized

in the following equation:

B(Z,A) = C1A− C2A2/3 − C3

Z2

A1/3(1− C4

C3

A−2/3)

− C1k(N − Z)2

A(1− C2

C1

A−1/3) + δ (1.6)

The terms are: the volume term (bulk binding), surface correction, Coulomb term with

surface diffuseness correction, the asymmetry term with surface correction, and the pairing

term δ. The Ci coefficients are fit to the empirical binding energies of nuclei. This phe-

nomenological model does a remarkable job of reproducing experimental binding energies

across the nuclear chart, Fig. 1.4. However, at certain values of neutron number, N, one can

see large deviations with irregular spacing from the liquid drop model, suggesting a need for

a better treatment of the quantum mechanical nature of nuclei.

10

Figure 1.5: Single-particle levels for the IPM with (from left to right) (a) the harmonicoscillator, (b) infinite square well, (c) finite square well and (d) Wood-Saxon potentials. Thefinal column is for a Wood-Saxon potential with the spin-orbit term included.

11

The most common quantum mechanical approach to the many-body nuclear problem

is the independent particle model (IPM). It is a mean-field theory in which the nucleus is

approximated by considering each nucleon as an independent particle moving in an average

potential well created by all of the other nucleons. This simplifies the N-body problem into

N, one-body problems. The result is a shell structure with single-particle levels, completely

analogous to what is taught about the electronic structure of atoms in introductory classes.

The form of the potential changes the ordering of the levels in energy, with the best choice

for nuclei being a square well with rounded edges (Wood-Saxon) potential. The potential

well for protons includes a Coulomb interaction in addition to the central potential felt by all

nucleons. The addition of spin-orbing (ℓ·s) coupling to the Wood-Saxon potential reproduces

the shell gaps seen experimentally for nuclei, Fig. 1.5. The ordering of these levels works well

for light (A<50) nuclei near stability, but some reordering of the levels has been observed in

some exotic nuclei.

1.3.2 Isospin

In nuclear physics it is often useful to introduce the quantity known as isospin, t or T , which

is an approximate quantum number related to the strong force. This quantum number

follows all of the same algebraic rules as spin or angular momentum. A generic nucleon has

t = 1/2, with the isospin projection tZ = −1/2 for a proton and tZ = 1/2 for a neutron.

This definition is arbitrary and different subfields choose different conventions. Using this

convention we can define the total isospin projection of a given nucleus as,

TZ =(N − Z)

2(1.7)

where N is the number of neutrons and Z is the number of protons. Most nuclear states

are characterized by a single isospin, a quantum number with a value greater or equal to its

projection. A few states have mixed isospin character which reinforces the fact that isospin

12

Figure 1.6: Isobar diagram for A=7. Levels are labeled by their energy, spin, and isospinwhen known, with intrinsic widths indicated by the size of the cross-hatching. States withidentical quantum numbers are connected by dashed lines to their analogs. Thresholds forparticle decays are labeled with their energy relative to the ground state of the given nucleus.Figure taken from Ref [1].

13

is only an approximate quantum number. The lowest lying states have the lowest values

of isospin (equal to the magnitude of its projection) and with increasing excitation energy,

progressive thresholds are passed at which higher isospin states are allowed.

For a given isospin T , there are 2T+1 projections (referred to as an isospin multiplet),

each corresponding to a state in a different nucleus. An example of this can be seen in Fig.

1.6. The ground state of 7He has spin and parity, Jπ, of 3/2−. The isospin projection is

TZ = 5−22

= 32, and the total isospin is T = |3

2| = 3

2. Following the dotted lines along the

isobaric chain one can see that in each nucleus one finds a state with the same Jπ and T as

the 7He ground state. These special excited states in 7Li and 7Be are the lowest T = 3/2

states in those nuclei and are known as isobaric analog states (IAS). They have all of the

same nuclear quantum numbers as their analogs (the ground states of 7He and 7B) with the

exception of the projection of the isospin quantum number. Chapters 3, 6, and 7 will focus

on these special states.

1.3.3 Proton Decay

Beyond the proton dripline, the proton separation energy becomes negative (energy is re-

leased by removing a proton). These proton-unbound nuclei serve as impediments to the

rp-process. In general, if element EAZ is proton unbound, when element EA−1

Z−1 tries to cap-

ture a proton to form EAZ , E

AZ will decay back to EA−1

Z−1 + p, and the rp-process will not

proceed until EA−1Z−1 beta decays. For some heavy nuclei, where the proton-decay half-life is

sufficiently long, proton-unbound element EAZ can capture a second proton to “skip over”

this rp-process waiting point.

Thermodynamically, once a nucleus has a negative proton-separation energy, it will one-

proton (1p) decay. However, kinetically it can compete with beta decay. The Coulomb

barrier for proton decay increases with increasing Z. For a fixed decay energy and angular

momentum, this means the barrier that the proton must penetrate through becomes higher

and thicker, and the half-life of the state increases. If it increases sufficiently, beta decay

14

Figure 1.7: Partial levels schemes for the different classifications of 2p decay. Direct 2p decayis shown in (a) and (c), sequential in (b) and democratic in (d,e). Figure taken from Ref [2].

can become a competitive decay mode. This is true for the first discovered proton emitter,

151Lu [11], where only 60% of the total decay is via proton decay.

Two-proton (2p) decay was first proposed by Goldansky in the 1960s for even-Z nuclei

[12]. He predicted that nuclei near the proton dripline could undergo 2p decay when the

pairing interaction made the 1p decay energetically forbidden, see Fig. 1.7 (c). This was

the case for the discovery of the first 2p decaying nucleus, 6Be [13]. A nucleus could in

principle 2p decay if the only energetically accessible 1p intermediate is lower in energy than

the 2p threshold, Fig. 1.7 (a), however this will only occur if the 1p channel is suppressed by

nuclear structure. When the 1p intermediate is energy accessible and above the 2p threshold,

a nucleus can 2p decay through two sequential steps of 1p decay, Fig. 1.7 (b). If the 1p

intermediate is wide relative to the decay energy, Fig. 1.7 (d,e), then the distinction between

direct and sequential 2p decay becomes less clear. In these cases, often called democratic

decay, the lifetime of the 1p is sufficiently short that it has no effect on the kinematics of

the decay. To any experiment, these decays appear the same as direct 2p decay. All three of

these classifications are based entirely on energetics. Chapter 3 will expand these cases and

classifications to include the effects of isospin.

15

Figure 1.8: Jacobi vectors for three particles in coordinate and momentum spaces in the “T”and “Y” Jacobi system. The X and Y vectors define each system. Figure taken from Ref[3].

1.3.4 Decay Correlations

In a three-body decay (like 2p decay), there are nine degrees-of-freedom needed to describe

the momenta of the particles. Three of these are associated with the center-of-mass motion,

and can be removed by working in the center-of-mass frame. Three more describe the Euler

rotation of the decay plane. For a spin 0 system, these rotation angles can also be ignored.

However, if the system has a finite spin these angles contain information about spin alignment

of the system. In this work, we will not worry about any alignment as it plays no role on

the internal degrees of freedom. One final degree-of-freedom can be constrained because the

decay energy is fixed. This leaves only two degrees-of-freedom associated with the relative

motion of the three-body system. Two common choices, the Jacobi “T” and “Y” systems,

are shown in Fig. 1.8.

16

The energy and angle parameters for the Jacobi momenta, kx, ky, can be described by:

ε = Ex/ET , cos(θk) = (kx · ky)/(kx ky) ,

kx =A2k1 − A1k2

A1 + A2

, ky =A3(k1 + k2)− (A1 + A2)k3

A1 + A2 + A3

,

ET = Ex + Ey = k2x/2Mx + k2

y/2My, (1.8)

where Mx and My are the reduced masses of the X and Y subsystems, and Ai are the mass

numbers of the particles. For the Jacobi “T” system, k3 is assigned to be the core. Then

the energy parameter, Epp, is the relative energy between the two protons. For the Jacobi

“Y” system, k3 → kp, and the energy parameter describes the relative energy between the

core and the proton, Ecore−p. The angular parameter, in both systems, describes the angle

between kx, which is the momentum of particle 1 in the center-of-mass frame of particles 1

and 2, and ky, which is the center-of-mass momentum of particles 1 and 2 in the center-of-

mass frame of the whole system. For example, in the Jacobi “Y” frame if the two protons

are emitted back-to-back (opposite sides of the core), then cos(θk) = 1. If they are emitted

close together, cos(θk) ≃ −1. These are referred to colloquially as “cigar” and “diproton”

configurations.

17

Chapter 2

Experimental Methods

2.1 Overview

There were two separate, but related, goals that motivated the experiments of this thesis.

The first goal was to measure the decay energy, intrinsic width, decay mode, and decay

correlations for the isobaric analog state (IAS) of 8C ground state (g.s.) in 8B. A previous

study [14] found that the ground state of 8C decays via two sequential steps of prompt 2p

decay (through 6Beg.s.). As that experiment was designed to detect the decay products of

8C, namely protons and alphas, the ranges for the silicon ∆E detectors were not set up to

measure particles heavier than helium. In a few detectors, a small amount of 6Li was seen

that did not over-range the amplifiers. In the invariant-mass spectrum of 2p+6Li there was

a peak at 7.05 MeV. This could either correspond to an unknown state in 8B at 7.05 MeV

if the 2p decay directly populated the 6Lig.s., or to a state at 10.61 MeV if it decayed to

the excited state at 3.5 MeV in 6Li. This state is the T=1 isobaric analog, the analog of

6He, and γ decays to the ground state. As the previous experiment was not sensitive to γ

rays, they could not determine this directly, however 10.61 MeV was the known excitation

energy of the isobaric analog state (IAS) of 8B, but it was not known how this continuum

state decayed. Thus a new experiment was performed with a) higher ranges on the silicon

18

detectors to measure the 6Li fragments without bias and b) an array of gamma detectors

was added around the target to detect any γ rays in coincidence with the 2p decays.

The second goal of these experiments was to measure a second ground-state/ isobaric

analog pair, 16Negs and 16FIAS. While the ground state of 16Ne had been seen in transfer

reactions in previous studies [15, 16, 17, 18] and this state was known to be unstable with

respect to 2p decay, no data existed on the three-body decay correlations. The measured

intrinsic width was far wider than theoretical predictions and our experiment would remea-

sure the width with better resolution than the previous work. Nothing was previously known

about the isobaric analog state in 16F, however it was expected to decay by 2p emission to

the isobaric analog state in 14N. This is analogous to the decay of 8BIAS.

These experiments were performed at Michigan State University at the National Super-

conducting Cyclotron Laboratory. A schematic of the experimental setup is shown in Fig.

2.1. Using a set of coupled cyclotrons (K500 + K1200) a beam of 16O nuclei, accelerated to

150 MeV/A (roughly half the speed of light), was impinged on a 9Be target. The beam un-

derwent reactions with the target and produced numerous light fragments. Using a magnetic

separator, these fragments were filtered and a secondary beam of 9C (68 MeV/A, 9.0×104

pps and 50% purity) was produced. This beam was transported to the secondary reaction

target where the 8BIAS was produced by proton knockout reactions. For the second experi-

ment a primary beam of 20Ne (170 MeV/A) was used to produce a secondary beam of 17Ne

(63 MeV/A, 1.2×105 pps and 11% purity). Then states in 16Ne and 16F were populated

following neutron and proton knockout respectively.

2.2 Invariant-mass Method

For particle-decaying states with lifetimes shorter than ≈ 10 ns, the invariant-mass method

is ideal for measuring the decay energy and, in the case of three-body decay, the momentum

correlation between the fragments. The basic idea of the method is to measure the momen-

19

Figure 2.1: Schematic of the experimental setup at Michigan State University. Stable beams(green) are accelerated first by the K500 and then the K1200 cyclotrons. The stable beamimpinges on a 9Be target, where it is fragmented. The fragments are separated by the A1900magnetic fragment separator based on their mass-to-charge ratio. These radioactive beams(red) are transported to the experimental area where they undergo further reaction to createthe unbound products. Charged particles are detected with HiRA, and γ rays are detectedby CAESAR.

20

tum vectors and particle type for all of the products of the decay and use that information

to reconstruct the mass of the parent state in the rest frame of that particle. The invariant

mass (in units of energy) is given by Eq. 2.1

Mc2 =

√(∑i

Ei)2)− (∑i

pic2)2 (2.1)

where Ei and pi are the total energy and momentum of each fragment. In these experiments

the momentum of each particle is determined by measuring the energy deposited in the

silicon and CsI(Tl) detectors of HiRA with the direction of each vector determined from the

x and y strips of the silicon detector and assuming the reaction took place in the center

of the target and the mass extracted using the standard ∆E − E technique. This method

relies on the fact that different particles deposit their energy at different rates based on their

stopping power, Eq. 2.2

dE

dx∝ Z2A

E(2.2)

where Z is the ion’s charge, A is the ion’s mass number, and E is the kinetic energy (At

the energies relevant here, light nuclei are fully stripped and thus their charge q = Z.). By

plotting the energy lost in a thin transmission detector (∆E) vs the energy deposited in

a thick stopping detector (E), the ions will separate into groups of bands with the same

charge, Z, with each band corresponding to a different isotope, Fig. 2.2.

2.3 The Detectors

The charged particles produced in the decay of the nucleon knockout products were detected

in the High Resolution Array (HiRA) [19]. For these experiments, HiRA consisted of 14 Si-

CsI(Tl) ∆E − E telescopes located 85 cm downstream of the secondary target, subtending

polar angles from 2.0 to 13.9 in the lab. The telescopes were arranged in five towers with

21

CsI Light [arb.]0 500 1000 1500 2000

[M

eV]

Si

E∆

0

10

20

30

40

50

60

70

pd t

He3 α

Li6Li7

Be7

Figure 2.2: Energy deposited in a silicon (∆E) vs energy deposited in a CsI(Tl) (E) for atypical outer HiRA telescope. The silicon energy has been calibrated and the CsI(Tl) energyis uncalibrated. Different bands are labeled by their corresponding isotope.

22

Figure 2.3: HiRA Array viewed from upstream.

23

Figure 2.4: Layout of CAESAR detectors. (left) cross-sectional view perpendicular to beamaxis showing rings B and F and (right) cross-sectional view parallel to beam axis showingall ten rings and target position (dot with vertical line through it). Taken from Ref. [4].

a 2-3-4-3-2 arrangement, Fig. 2.3. The center tower had a small gap between the two inner-

most telescopes, centered on the beam axis, to allow for the unreacted beam to pass through.

Each telescope consisted of a 1.5-mm-thick double-sided silicon strip ∆E detector followed by

a 4-cm-thick CsI(Tl) E detector. Each of the 14 HiRA telescopes has four CsI(Tl) detectors,

each spanning a quadrant of the preceding Si ∆E detector. Signals produced in the Si

were processed in one of two ways. For the two detectors immediately above and below the

beam, the signals were amplified using external charge-sensitive amplifiers (CSAs) and then

resistively split into low- and high-gain channels before being processed by the HINP16C

ASIC (application-specific integrated circuit) electronics previously designed by our group

[20]. This provided roughly six times the dynamic range of the other telescopes. The other

Si detectors were processed with the HINP16C chip electronics and amplified with CSAs

internal to the chip. Signals from the CsI(Tl) detectors were processed using conventional

electronics.

γ rays were measured in coincidence with charged particles using the CAEsium-iodide

24

scintillator ARray (CAESAR) Ref. [4], Fig. 2.4. In these experiments CAESAR comprised

158 CsI(Na) crystals covering the polar angles between 57.5 and 142.4 in the laboratory

with complete coverage in the azimuthal angles. The first ring (A) and the last two rings (I,J)

of the full CAESAR array were removed due to space constraints. Ring B consisted of 10,

3” x 3” x 3” crystals and rings C-H consisted of 24, 2” x 2” x 4” crystals in a closely-packed

geometry. Signals from the CsI(Na) detectors were processed using conventional electronics.

2.3.1 Silicon

The High Resolution Array (HiRA) is comprised of 14 “stack” telescopes, consisting on a

thin transmission detector and a thick stopping detector. The thin detectors of HiRA in

this configuration are ion-implanted passivated silicon semiconductor detectors. The silicon

detectors, manufactured by Micron Semiconductor [21], are 6.4 cm x 6.4 cm x 1.5-mm-thick

with the faces subdivided into 32 strips orthogonal to one another on the front and back.

This high segmentation provides good angular resolution (∆θ/θ = 0.1) and allows multiple

particles to be detected in the same detector. The reduced area for each strip also reduces

the capacitance of each detector element thus reducing the noise.

Semiconductor detectors work on the following principle. As a charged particle moves

through the detector volume, electron-hole pairs are created. The number of pairs created

depends on the stopping power, dE/dx, which determines the energy deposited (Ed), and

the band gap of the material, which determines the amount of energy needed to create an

electron-hole pair. For silicon the average energy required to create a electron-hole pair is

3.7 eV, therefore Ed = 1 MeV can produce 106/3.7 ≈ 270,000 pairs. Normally these pairs

will quickly recombine, however by applying a modest electric field (E ≈ 2000 V/cm for a

HiRA detector) the electrons and holes will drift towards opposite surfaces where they can

be collected. Since the production of pairs is linearly proportional to the energy deposited,

these detectors can be used to measure charged particles over a large range of energies. A

standard silicon detector is made of a sandwich of two silicon crystals with different dopants

25

to make a p-n junction. N-type silicon is made by adding a group V element as an impurity

into the bulk and p-type is made by adding group III. In a p-n junction, thermal diffusion will

drive electrons from the n-type (donor) region to the p-type (acceptor) region. This leads to

a region in the middle which is depleted of charge carriers, which creates a barrier to current

flow. By applying a positive potential to the n region and negative to the p region (often

called reverse biasing) this barrier across the junction can be enhanced. A silicon detector

is a p-n junction that has been reversed biased to the point that the bulk of the detector is

“fully depleted” of charge carriers. When a charged particle creates electron-hole pairs, they

are swept to the respective surfaces and collected through metal contacts deposited on the

surfaces. For a HiRA detector the total energy deposited can be determined through either

the electrons or holes collected and the position of the deposited energy can be determined

by which strip collects charge on the front and back.

2.3.2 Scintillators

The thick “stopping” detectors in a HiRA telescope are 4-cm-long CsI trapezoidal crystals

doped with thallium at the ppt level. These crystals are only slightly hygroscopic, which

makes them ideal for applications for which they cannot be placed into air-tight cans, such as

charged-particle detection (A can would absorb part of the energy, leading to worse energy

resolution). These crystals are 3.5 cm x 3.5 cm in the front and 3.9 cm x 3.9 cm in the back.

Behind each CsI(Tl) is a 3.9 cm x 3.9 cm x 1.3 cm light guide optically coupled by BC 600

optical cement. A 1.8 cm x 1.8 cm silicon photodiode was glued to each light guide with

RTV615 silicon rubber. The light output of the CsI(Tl) crystal is well suited in wavelength

for photodiode readout. To make each crystal optically separated, the crystal is wrapped

in cellulose nitrate membrane filter paper on the long sides, with thin aluminized Mylar foil

covering the front. The light guides are painted with BC-620 reflective paint. Four crystals

are wrapped together with Teflon tape and placed behind a silicon detector.

Unlike a silicon detector, where energy is measured by the number of electron-hole pairs

26

produced, scintillators like CsI(Tl) measure energy by the light produced. When a charged

particle enters a crystal, ionization creates unbound electron-hole pairs or bound electron-

hole pairs called excitons. These excitons move freely within a crystal until they are trapped

by impurities (sites of dopant atoms) or crystal defects. Once the exciton is trapped, it

will radiatively de-excite with the emission of a visible photon. This light is collected by the

photodiode where it creates electron-hole pairs which are collected and read out as a current.

The excitons can also non-radiatively de-excite, the energy then goes into heat (crystal

vibration). The non-radiative channels represent about 75 % of the total de-excitation

channels.

CsI doped with sodium has about twice the light output of CsI doped with thallium. This

comes at the cost of being highly hygroscopic, so it must be hermetically sealed in a can. The

CsI(Na) detectors of the CAESAR array work the same way as CsI(Tl). γ rays can deposit

all of their energy into the kinetic energy of a single electron via the photoelectric effect or

part of their energy via Compton scattering. This electron further ionizes the medium, and

the electron-hole pairs produced de-excite as in CsI(Tl). If the γ ray has energy larger than

1.022 MeV, it can also undergo pair production and spontaneously create a positron-electron

pair in the presence of matter. These charged particles lose energy as before, but when the

positron is thermalized it will annihilate with an electron and create two 511 keV γ rays.

One or both of the annihilation photons can escape the detector reducing the deposited

energy, yielding single and double escape peaks. The Compton scattered γ ray and those

created from the positron annihilation, if detected in a neighboring crystal, can be added to

the energy collected in the crystal with the first interaction.

2.3.3 Calibration

In order to determine the total energy deposited by the charged particles, accurate energy

calibrations are needed. The silicon detectors were calibrated using a 228Th multi-line α

source. 228Th is part of the natural decay chain beginning with 232Th and will decay with

27

Energy [MeV]4 5 6 7 8 9 10

Co

un

ts

0

50

100

150

200

250

Figure 2.5: Calibrated alpha energy spectrum for a typical silicon strip in the front (holecollecting) side. The FWHM of the 8.785 MeV peak is 72 keV.

28

Parent t1/2 Eα0 Eα1 fα0 fα1 fβ(MeV) (MeV)

228Th 1.912 y 5.423 5.340 0.726 0.274 0224Ra 3.660 d 5.685 5.449 0.949 0.051 0220Rn 55.60 s 6.288 – 1 0 0216Po 0.145 s 6.778 – 1 0 0212Pb 10.64 h – – 0 0 1212Bi 60.55 m 6.090 6.051 0.098 0.254 0.648212Po 0.299 µs 8.785 – 1 0 0208T l 3.053 m – – 0 0 1208Pb stable – – – – –

Table 2.1: Alpha energies, half-lives, and branching ratios for the 228Th decay chain. α′swith less than 5% intensity are not included in this table. α0 and α1 denote decay to theground state and first-excited state of the daughter, respectively.

5 α′s and 2 β′s to 208Pb, with the energies and branching ratios fα,β listed in Table 2.1. A

fingerprint of the known energies and intensities can be fit to the measured alpha energy

spectrum, from which a linear calibration of the ADC channel number to energy can be

extracted. A typical calibrated alpha energy spectrum for a HiRA silicon can be found

in Fig. 2.5. While the charge output is linear with deposited energy, the electronics can

have non-linearities. To test for this, test pulses of a known voltage were injected into the

electronics. A spectrum of these pulses can be seen in Fig. 2.6. The electronics were found

to be linear at low pulse heights with a compression (more energy per channel) at high pulse

height.

Source Eγ Iγ(MeV)

Am-Be 4.438 160Co 1.332 0.9998

1.173 0.998522Na 0.511 Annihilation

1.2744 0.999488Y 0.898 0.937

1.836 0.992

Table 2.2: γ-ray energies (Eγ) and gamma fraction per decay (Iγ) for the sources used tocalibrate CAESAR. γ rays from the Am-Be source come from the 9Be(α,γ)12C reaction.

29

Energy [Channel Number]0 5000 10000 15000

Co

un

ts

0

200

400

600

800

1000

1200

1400

Figure 2.6: Pulser calibration spectrum for a typical silicon strip on the front side. Thepulses set in uniform charge increments and a non-linearity of the electronics above halfscale is clearly seen. The pulse corresponding to the peak near channel number 10,000 wasrun for three times as long as the other pulses.

30

The light output of a CsI(Tl) scintillator is highly dependent on the charge and mass of

the incident ion. For a given energy, a heavy ion will deposit more energy into a medium

than a light particle, from Eq. 2.2. With a high dEdx, the high density of ionization leads

to a quenching of the light. This means that the energy calibration of the CsI(Tl) will

be particle dependent. Calibration is accomplished by the use of two “cocktail” beams of

different magnetic rigidity containing all of the isotopes of interest. For these experiments we

had two calibration beams with energies 55 MeV/A and 75 MeV/A for the N=Z nuclei. Due

to the momentum acceptance of the magnetic separator used to filter the beams, separate

beams of the same energy containing only protons were needed to calibrate the protons.

The CsI(Na) detectors of the CAESAR array were calibrated using a standard set of γ-ray

calibration sources. These sources and their γ-ray energies are listed in Table 2.2.

31

Chapter 3

Isobaric Analog State of 8B

3.1 Background

Proton-rich nuclei beyond the proton drip line will decay by the emission of charged particles.

In some cases these nuclei will decay by emitting two protons in a single step, i.e. 2p decay.

In a recent paper it was found that the ground state of 8C, the mirror of the 4-neutron halo

system 8He, has a very unusual decay [14]. It decays by 2p emission to the ground state of

6Be, the mirror of another neutron halo system 6He. The 6Be nucleus itself undergoes 2p

decay, and thus 8C undergoes two sequential steps of 2p decay. In this chapter I will examine

the decay of the isobaric analog state (IAS) of 8C in 8B. In Ref. [14] our group presented

evidence that this state also undergoes 2p decay to the isobaric analog state of 6Be in 6Li.

This would be the first case of 2p decay between IAS states and this decay would be the

analog of the first 2p-decay step of 8Cg.s., in both cases the 2p decay is between T=2 and

T=1 states.

Prompt two-proton emission was originally thought to occur only when one-proton decay

was energetically forbidden [12]. However this definition has been extended to democratic

2p decay, see Fig. 1.7 (d,e), where 1p decay is energetically allowed but where the 1p decay

energy is of the same magnitude as the width of the 1p daughter [22]. The confirmation of

32

the 2p decay of the 8BIAS would further extend this to a third class of 2p emitters where 1p

decay is energetically allowed, but isospin forbidden.

With increasing mass, 12O is the next known 2p emitter after 8C. Its isobaric analog

state in 12N has also been ascribed to this new third class of 2p emitters [23] decaying to the

isobaric analog state in 10B. In both 8BIAS and 12NIAS decay, the daughter isobaric analog

states decay by γ emission, but in both studies [14, 23] the γ ray was not detected allowing

some uncertainty to the interpretation of the decay sequence and thus in the existence of

this third class of 2p decay. This deficiency is remedied in the present work where the 2p

decay of 8BIAS is revisited and the γ ray from the decay of the 6LiIAS daughter is observed.

In addition we measure the correlations between the decay products and compare them to

those previously determined for the 2p decay of 8Cg.s..

3.2 Decay mode

From the decay scheme shown in Fig. 3.1 one can see that one-nucleon decay from the IAS in

8B is either energy allowed but isospin forbidden (p), or isospin allowed but energy forbidden

(n). Two-proton decays to the ground and first excited states of 6Li are also energy allowed

but isospin forbidden. The only energy and isospin allowed decay mode is 2p decay to the

Jπ = 0+, T = 1 IAS in 6Li which is known to decay to the ground state of 6Li by emitting

a 3.563-MeV γ ray [9].

The reconstructed excitation-energy distribution of 8B fragments from detected 2p +

6Li events is shown in Fig. 3.2(a). The excitation energy was calculated based on the

assumption that the detected 6Li fragment was produced in its ground state. The only

state in 6Li with any significant gamma-decay branch is the IAS (T=1) state at 3.563-MeV.

The narrow peak at 7.06±0.020 MeV from the reconstructed particle energies was observed

previously [14] and was assigned as the IAS in 8B at an excitation energy of 10.61 MeV

assuming the decay populated the IAS in 6Li. The spectrum of γ rays in coincidence with

33

Ene

rgy

[MeV

]

0

5

10

15

g.s.B8

)+(T=1,2

IAS+0(T=2)g.s.B7n+

(T=3/2)

g.s.Be7p+)-(T=1/2,3/2

-5/2 -7/2

IAS-3/2(T=3/2)

g.s.Li62p+(T=0)

+dα

IAS+0(T=1)

Figure 3.1: Set of levels relevant for the decay of the IAS in 8B. The levels are labeled bytheir spin-parity (Jπ) and isospin (T) quantum numbers. Colors indicate isospin allowedtransitions. The width of the Jπ = 7/2− state in 7Be is not known, but assumed to be wideas in the mirror.

34

Co

un

ts

0

200

400

600

800

Li6 2p + →B 8

G1

a)

[MeV]ParticleE*5 6 7 8 9 10 11 12 13 14

Co

un

ts

0

20

40

α 2p + d + →B 8

b)

Figure 3.2: The excitation-energy spectrum from charged-particle reconstruction determinedfor 8B from detected a) 2p-6Li and b) 2p-α-d events.These energies assume no missing decayenergy due to γ-ray emission.

35

[MeV]γE0 1 2 3 4

Co

un

ts

0

50

100

150

3.56 MeVFirst Escape

Figure 3.3: Gamma-ray energies measured in coincidence with p-p-6Li events. This spectrumincludes add-back from nearest neighbors and has been Doppler-corrected eventwise. Thepeak at about 3 MeV is when one of the two 511 keV γ rays from the annihilation of thepair produced positron escapes detection in CAESAR.

36

the reconstructed 2p+6Li events satisfying gate G1 in Fig. 3.2(a) is shown in Fig. 3.3.

The γ-ray energies are corrected for nearest neighbor scattering and are Doppler-corrected

eventwise. The spectrum has two main peaks, one at 3.56 MeV and another 511 keV lower.

As a 3.563 MeV γ ray has a high probability for pair-production, a large fraction of time

the γ ray will produce a positron-electron pair in one of the detectors. The charged particles

will lose energy through interactions with the electrons in the crystal, depositing the bulk

of their energy into that detector. When the positron slows down sufficiently, it can bind

with an electron in the crystal forming positronium. The positronium will spiral in and

annihilate the positron and electron with the production of two 511 keV γ rays. When one

of these two γ rays is not detected in the array, the peak around 3 MeV is formed. The

observed spectrum is consistent with a single gamma ray with an energy of 3.563 MeV with

a significant single escape probability, confirming the IAS-to-IAS decay path. Adding the

3.563 MeV γ ray energy to the centroid of the peak in Fig. 3.2(a), gives us a total excitation

energy of 10.614±0.020MeV which is consistent with the tabulated value of 10.619±0.009

MeV[9].

The emission of a γ ray from the 6Li fragment will cause the fragment to recoil. If the

particle was at rest, this effect would be tiny and could be immediately ignored. Since the

fragment is moving and has a low mass, the recoil could actually have a measurable effect.

This was tested using a Monte Carlo simulation, and the change in the centroid from the

recoil is 0.002%, well within statistical uncertainties. The measured width changed by 1%,

which is small compared to the other uncertainties in the detection system.

While the γ ray following the 2p decay of 12NIAS was not measured in Ref. [23], the decay

scheme is logically the same as for 8BIAS. The present measurement for 8BIAS therefore

provides support for the assigned decay path for 12NIAS [23].

In light nuclei, isospin violation at the few percent level is not uncommon. Therefore it is

not surprising that we also see weak decay branches from 8BIAS to the low lying T=0 levels in

6Li. From the correlations between the decay products, the 2p+d+α exit channel is studied

37

as well, Fig 3.2(b). The channel is populated mostly by an isospin-forbidden 2p decay to the

Jπ=3+ excited state of 6Li that subsequently decays to a d+α pair. There is also a hint of

2p decay directly to 6Lig.s., Fig. 3.2(a). After correction for detector efficiences, the decays

through these channels have yields no more than: 10% (2p+d+α) and 11% (2p+6Lig.s.)

relative to the isospin-conserving decay. No evidence was observed for isospin-forbidden

decay to the p+7Be channel. At the 3σ level, we deduce an upper limit of 7.5% for this

decay.

3.3 Correlations

I now turn to the three-body correlations in the 2p decays of 8BIAS and its isospin partner

8Cg.s.. The energy and angular correlations in both the Jacobi T and Y coordinates are

shown in Fig. 3.4 for 8BIAS decay (left) and the first step of 8Cg.s. decay (right). In order to

determine the three-body correlations in the first step of 8Cg.s. decay, we required that one

and only one of the six possible pairs of protons, together with the α particle, reconstructed

to the correct invariant mass of the 6Be intermediate [14]. This event selection places some

uncertainty on the extracted correlations due to a background of misassigned pairs of protons

from the first and second 2p-decay steps. This background is expected to be smooth, and one

estimation of this background, described in Refs.[14, 24] is shown by the dashed curves in Fig.

3.4(e-h). In the Jacobi T energy distribution for 8Cg.s. decay, Fig. 3.4(e), an enhancement

at low relative proton energies, a region often called the “diproton” region, is observed. The

Jacobi T energy distributions for the decay of the isospin partner 8BIAS are shown in Fig.

3.4(a). Distortions due to the detector acceptance and resolution are expected to be small

and of similar magnitude for 2p decay of 6Be and 8C [14, 25]. In both cases one observes two

broad features at low and high relative energies, corresponding to so called “diproton” and

“cigar” configurations. However, the enhancement of the diproton region seen for 8C decay,

is not observed for 8BIAS decay.

38

Cou

nts

0

50

100

150

200

250

"T"a)

IASB8 Cou

nts

0

20

40

60

80

100

"T"e)

g.s.C8

T/ExE0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cou

nts

0

200

400

600

800

"Y"b)

T/ExE0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cou

nts

0

50

100

150

200

250

"Y"f)

Cou

nts

0

200

400

600

800

"T"c)

Cou

nts

0

50

100

150

200

250

"T"g)

)kθcos(−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Cou

nts

0

200

400

600 "Y"d)

)kθcos(−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Cou

nts

0

100

200"Y"h)

Figure 3.4: Left panel (a-d) are the projected three-body correlations from the decay of8BIAS to the 2p+6LiIAS exit channel in the (a)(c) T and (b)(d) Y Jacobi systems. Energycorrelations are shown in (a) and (b), angular correlations in (c) and (d). The right panelshows the same as the left, but now the correlations are associated with the first step of 8Cg.s

decay to 2p+6Be.

39

The Jacobi Y energy distributions for 8BIAS decay and 8Cg.s. decay are shown in Fig.

3.4(b) and (f) respectively. In a three-body decay, the two protons should have approximately

equal energies as this maximizes the product of their barrier penetration factors [12]. This

is evidenced by the observation of a single peak at Ex/ET=0.5 in the proton-core relative-

energy spectrum (Jacobi Y system) and suggests that these are prompt 2p decays. As

was seen for 6Beg.s, the Jacobi Y Ex/ET and Jacobi T θk distributions contain the same

information, complementary to the Jacobi Y θk and Jacobi T Ex/ET plots.

3.4 Summary

We have confirmed that IAS-to-IAS 2p decay can become the dominate decay mode when

all one-nucleon emission channels are either energy or isospin forbidden. The three-body

correlations from the two-proton decay of 8BIAS to 6LiIAS were measured and found to be

statistically different from its isospin partner 8Cg.s.. The origin, or origins, of this difference

is uncertain. The difference could result from distorting effects of the long-range Coulomb

interaction which must be followed out to tens of thousands of fm in 2p decay theory [26].

While both decay initially to three charged particles, the 6Be from the decay of 8Cg.s. will

further decay to 2p+α within a few hundred fm of the initial decay, well within the range

of the distorting final-state Coulomb interaction. Thus it is possible that the ultimate five-

body final state distorts the measured correlations between the reconstructed fragments of

the first three-body decay of 8Cg.s., a distortion that would not be present for 8BIAS. However

the difference could also arise from the much shorter lifetime of 8Cg.s. (τ = 2.5x10−21 s) as

comparted to 8BIAS (τ > 5.5x10−21 s), which would allow the former to couple more strongly

to the entrance channel knockout production reaction [27].

40

Chapter 4

16Ne Ground State

4.1 Background

Two-proton (2p) radioactivity [28] is the most recently discovered type of radioactive decay.

It is a facet of a broader three-body decay phenomenon actively investigated within the

last decade [2]. In binary decay, the correlations between the momenta of the two decay

products are entirely constrained by energy and momentum conservation. In contrast for

three-body decay, the interfragment correlations are not constrained by conservation laws

and are sensitive to the decay dynamics and perhaps also sensitive to the internal nuclear

structure of the decaying system. In 2p decay, as the separation between the decay products

becomes greater than the range of the nuclear interaction, the subsequent modification of

the initial correlations is determined solely by the Coulomb interaction between the decay

products. As the range of the Coulomb force is infinite, its long-range contribution to the

correlations can be substantial, especially, in heavy 2p emitters.

Prompt 2p decay is a subset of a more general phenomenon of three-body Coulomb

decay (TBCD) which exists in mathematical physics (as a formal solution of the 3 → 3

scattering of charged particles), in atomic physics (as a solution of the e → 3e process), and

in molecular physics (as exotic molecules composed from three charged constituents) [29, 30,

41

31, 32, 33, 34]. The theoretical treatment of TBCD is one of the oldest and most complicated

problems in physics because of the difficulty associated with the boundary conditions due

the the infinite range of the Coulomb force. The exact analytical boundary conditions for

this problem are unknown, but different approximations have been tried. In nuclear physics,

TBCD has not attracted much attention, however the three-body Coulomb aspect of 2p

decay will become increasingly important for heavier prospective 2p emitters [35].

Detailed experimental studies of the correlations have been made for the lightest p-shell

2p emitter 6Be [36, 37] where the Coulomb interactions are minute and their effects are

easily masked by the dynamics of the nuclear interactions [38]. The Coulomb effects should

be more prominent for the heaviest observed 2p emitters, however these cases are limited by

poor statistics; e.g. the latest results for the pf -shell 2p-emitters 54Zn [39] and 45Fe [40] are

based on just 7 and 75 events, respectively. Due to these limitations, previous 2p studies

dedicated to the long-range treatment of the three-body Coulomb interaction [41], found

consistency with the data, but no more.

The present work fills a gap between these previous studies by measuring correlations in

the 2p ground-state (g.s.) decay of the sd-shell nucleus 16Ne where the Coulombic effects

appear to be strong enough to be observable. Known experimentally for several decades

[42], 16Ne has remained poorly investigated with just a few experimental studies [17, 18, 15,

16]. However, interest has returned recently with the decay of 16Ne measured in relativistic

neutron-knockout reactions from a 17Ne beam [43, 44]. We study the same reaction, but at an

“intermediate” beam energy and obtain data with better resolution and smaller statistical

uncertainty. Combined with state-of-the-art calculations, we have identified the role the

long-range Coulomb interactions play in the measured three-body correlations.

Apart from the Coulomb interactions, predicted correlations show sensitivity to the initial

2p configuration and other final-state interactions previously identified in 2n decay [45, 46,

47]. Before the unambiguous statement can be made from the correlations related to short-

range nuclear interactions, the effects of the long-range Coulomb interactions must first be

42

isolated and quantified.

4.2 Excitation Spectrum

The spectrum of the total decay energy ET constructed from the invariant mass of detected

14O+p+p events is shown in Fig. 4.1. Due to a low-energy tail in the response function of the

Si ∆E detectors, there is leaking of a few 15O ions into the 14O gate in the ∆E−E spectrum.

However, this contamination can be accurately modeled by taking detected 15O+p+p events

and analyzing them as 14O+p+p. The resulting spectrum (dashed red curve) was normalized

to the ∼ 1-MeV peak associated with 2nd-excited state of 17Ne. All the other observed peaks

are associated with 16Ne, with the g.s. peak at ET = 1.466(20) MeV being the dominant

feature. This decay energy is consistent with the value of 1.466(45) MeV measured in [15] and

almost consistent with, but slightly larger than, other experimental values of 1.34(8) MeV

[17], 1.399(24) MeV [18], and 1.35(8) MeV [43], 1.388(14) MeV [44].

The predicted spectra in Fig. 4.1, see below for details, provide guidance for possible spin-

parity assignments of the other observed structures, suggesting that the previously known

peaks [43, 44] at ET = 3.16(2) and 7.60(4) MeV are both 2+ excited states. The broad

structure at ET ∼ 5.0(5) MeV is well described as a 1− “soft” excitation which is not a

resonance, but a continuum mode, sensitive to the reaction mechanism [37]. In the mirror

16C system, there are also J = 2(±), 3(+), and 4+ contributions in this energy range, but

for neutron-knockout from p1/2, p3/2, and s1/2 orbitals in17Ne, we should only expect strong

population for 0+ (p1/2 knockout), 2+ (p3/2 knockout) and 1− (s1/2 knockout) configurations.

I will concentrate on the g.s. for the remainder of this chapter (1.27 < ET < 1.72 MeV) and

all subsequent figures will show contamination-subtracted data.

43

4.3 Theoretical Model

The calculations used in this work were performed by Leonid Grigorenko and his group

in Dubna, Russia. They are the leading group in calculating nuclear three-body decays,

and the present study is the most recent joint effort between this theory group and our

experimental group. The model used in this work is similar to that applied previously to 16Ne

[48] but with improvements concerning basis convergence [3], TBCD [41] and the reaction

mechanism [36]. The three-body 14O+p+p continuum of 16Ne is described by the wave

function (WF) Ψ(+) with the outgoing asymptotic obtained by solving the inhomogeneous

three-body Schrodingier equation,

(H3 − ET )Ψ(+) = Φq,

with approximate boundary conditions of the three-body Coulomb problem. The three-body

part of the model is based on the hyperspherical harmonics method [3]. The differential cross

section is expressed via the flux j induced by the WF Ψ(+) on the remote surface S:

dσ

d3k14od3kp1d3kp2

∼ j = ⟨Ψ(+)|j|Ψ(+)⟩∣∣∣S. (4.1)

When comparing to the experimental data, the theoretical predictions were used in Monte-

Carlo (MC) simulations of the experiment [24, 36] to take into account the bias and resolution

of the apparatus.

The source function Φq was approximated assuming the sudden removal of a neutron

from the 15O core of 17Neg.s.,

Φq =

∫d3rne

iqrn⟨Ψ14O|Ψ17Ne⟩ , (4.2)

where rn is the radius vector of the removed neutron. The 17Neg.s. WF Ψ17Ne was obtained in

a three-body model of 15O+p+p that had been previously tested against various observables

44

[MeV]TE0 2 4 6 8 10 12

Cou

nts

1

10

210

310

410 O142p+Background

+2+0-1

1 1.5 20

200

400

Figure 4.1: Experimental spectrum of 16Ne decay energy ET reconstructed from detected14O+p+p events. The dashed histogram indicates the contamination from 15O+p+p events.The smooth curves are predictions (without detector resolution) for the indicated 16Ne states.The inset compares the contamination-subtracted data to the simulation of the g.s. peak forΓ = 0, ftar = 0.95, where the dotted line is the fitted background.

[49]. Similar ideas had been applied to different reactions populating the three-body contin-

uum of 6Be [36, 38, 37]. The 14O-p potential sets were taken from [48] which are consistent

with a more recent experiment [50], providing 1/2+ and 5/2+ states at Er = 1.45 and 2.8

MeV, respectively consistent with the experimental properties of these states in both 15F

and 15C. For the p-p channel a smooth local potential was used that fit two nucleon data up

to 300 MeV and reproduced properties of finite, closed shell nuclei [51].

The three-body Coulomb treatment in this model consists of two steps. (i) We are able to

impose approximate boundary conditions of TBCD on the hypersphere of very large (ρmax .

4000 fm) hyperradius by diagonalizing the Coulomb interaction on the finite hyperspherical

basis [52]. Within this limitation the procedure is exact, however it breaks down at larger

hyperradii as the accessible basis size become insufficient. (ii) Classical trajectories are

generated by a MC procedure at the hyperradius ρmax and propagated out to distances

ρext ≫ ρmax. The asymptotic momentum distributions are reconstructed from the set of

45

[MeV]TE1 1.5 2

Co

un

ts

0

100

200

300

400

500

600

Figure 4.2: Comparison of simulated line shapes of the best fit (red curve) and 3σ upperlimit (blue dashed curve) to the data. See text for description of fit parameters.

trajectories after the radial convergence is achieved. The accuracy of this approach has been

tested in calculations with simplified three-body Hamiltonians allowing exact semi-analytical

solutions [41].

4.4 Ground-state Width

The theoretical difficulty of reproducing the large experimental g.s. widths measured for 12O

and 16Ne has been pointed out many times in the last 24 years [53, 54, 48, 55, 56]. For 12O,

this issue was resolved when a new measurement [57] gave a small upper bound. For 16Ne,

previous measurements of Γ=200(100) keV [17], 110(40) keV [18], and 82(15) keV [44] are

large compared to the theoretical predictions, e.g. 0.8 keV in [48].

The experimental resolution is dominated by the effects of multiple scattering and energy

loss in the target. Their magnitudes were fined tuned in the MC simulations by reproducing

the experimental 15O+p+p invariant-mass peak associated with the narrow (predicted life-

time of 1.4 fs [58]) 2nd-excited state in 17Ne (see Chapter 8) by scaling the target thickness

46

from its known value by a factor ftar. The best fit is obtained with ffittar = 0.95 with 3-σ limits

of 0.91 and 1.00. With ffittar, we find that the simulated shape of the 16Neg.s. peak for Γ = 0

is consistent with the data (red curve in Fig. 4.2). To obtain a limit for Γ, we used a Breit-

Wigner line shape in our simulations and find a 3-σ upper limit of Γ < 80 keV with ftar = 0.91

(blue dashed curve in Fig. 4.2). This limit is the first experimental result consistent with

theoretical predictions of a small width [in the keV range, see, e.g. Fig. 4.4(a)]. However,

out limit is still considerably larger than the predictions, and on the other hand, it is still

consistent with two of the previous experiments so even higher resolution measurements are

needed to fully resolve this issue.

4.5 Three-body Energy-Angular Correlations

As described in the introduction, the internal final state coordinates of a three-body decay

can be completely described by two parameters: an energy parameter ε and an angle θk

between the Jacobi momenta kx, ky:

ε = Ex/ET , cos(θk) = (kx · ky)/(kx ky) ,

kx =A2k1 − A1k2

A1 + A2

, ky =A3(k1 + k2)− (A1 + A2)k3

A1 + A2 + A3

,

ET = Ex + Ey = k2x/2Mx + k2

y/2My, (4.3)

where Mx and My are the reduced masses of the X and Y subsystems. With the assignment

k3 → k14O, the correlations are obtained in the “T” Jacobi system where ε describes the

(fractional) relative energy Epp in the p-p channel. For k3 → kp, the correlations are obtained

in one of the “Y” Jacobi systems where ε describes the (fractional) relative energy Ecore-p in

the 14O-p channel.

The experimental and predicted (MC simulations) energy-angular distributions, in both

Jacobi representations are compared in Fig. 4.3 and found to be similar. More detailed

47

comparisons will be made with the projected energy distributions.

The convergence of three-body calculations is quite slow for some observables [3, 59].

Figure 4.4 demonstrates the convergence, with increasingKmax (maximum principle quantum

number of the hyperspherical harmonic method) for two observables for which the slowest

convergence is expected. These calculations provide considerable improvement compared to

the calculations of [48] which were limited to Kmax = 20.

To investigate the long-range nature of TBCD, we studied the effect of terminating

the Coulomb interaction at some hyperradius ρcut (That is beyond ρcut the particles just

free stream). The energy distribution in the “Y” Jacobi system is largely sensitive to just

the TBCD and the global properties of the system (ET , charges, separation energies) [2].

This makes it most suitable for studying the ρcut dependence [Fig. 4.5(a)]. Note that the

theoretical MC results are always normalized to the integral of the data. The comparison

with the data in Fig. 4.5(b) demonstrates consistency with the theoretical calculations only if

the considered range of the Coulomb interaction far exceeds 103 fm (ρcut = 105 fm guarantees

full convergence). This conclusion is only possible due to the high quality of the present data.

In contrast in [44], where the experimental width of the g.s. peak is almost twice as large and

its integrated yield is ∼ 3 times smaller, the corresponding ε distribution is broader with a

FWHM of 0.41 compared to our value of 0.33. This difference is similar to that obtained

over the range of ρcut considered in Fig. 4.5(a) demonstrating the need for high resolution to

isolate the distorting effects due to the Coulomb interaction at distances far exceeding those

at which the strong force can have an influence.

Our conclusions on TBCD are dependent on the stability of the predicted correlations

to the other inputs of the calculations. Figure 4.5(d) demonstrates the excellent stability

of the core-p energy distribution over a broad range (±200 keV) of ET . Indeed, in this

range we have a maximum in the width for this distribution due to the competition between

two trends: (i) the distribution tends to ε = 0.5 in the limit ET → 0 [28], and (ii) some

minimal width for the ε distribution is expected at ET ∼ 2Er, where Er ∼ 1.45 MeV is the

48

-1.0

-0.5

0.0

0.5

1.0

"T"

0+ Exp. MC0+

cos(

k )

0

20

40

60

80

100114

"T"

0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.5

0.0

0.5

1.0"Y" Exp. MC

= Epp (Ecore-p) /ET

cos(

k )

0

20

40

60

80

100112

0.0 0.2 0.4 0.6 0.8 1.0

"Y"

= Epp (Ecore-p) /ET

Figure 4.3: Energy-angular correlations for 16Neg.s.. Experimental and predicted (MC sim-ulations) correlations for Jacobi “T” and “Y” systems are compared.

49

0 20 40 60 80 1000

1

2

3

asy = 3.1 keV(16

Ne g

.s.)

(keV

)

Kmax

(a)

0.0 0.2 0.4 0.6 0.8 1.00

1

2 (b)

"T"

Kmax 20 40 60 80 100

dj/d

= Epp /ET

Figure 4.4: The convergence of the predicted (a) decay width and (b) energy distribution inthe “T” system on Kmax (maximum principal quantum number of hyperspherical harmonicsmethod). The asymptotic decay width of 16Ne assuming exponential Kmax convergence isgiven in (a) by the dashed line.

15Fg.s. → core + p decay energy. The predictions of such a “narrowing” [2] were recently

proven experimentally [36]. The curve for ET = 1.976 MeV is also provided in Fig. 4.5(d) to

show that a really large change in energy is required to produce a significant modification of

the ε distribution.

The other important stability issue is with respect to the properties of 15Fg.s. for which

there is no agreement on its centroid Er and width [60]. Figure 4.5(c) shows predicted

ε distributions based on four different 14O+p interactions which give the indicated 15Fg.s.

properties. Even if we use the data from [61], which differs the most from the other results

(Er ∼ 1.23 MeV instead of Er ∼ 1.4− 1.5 MeV), no drastic effect is seen.

The evolution of the energy distribution between the two protons with ρcut is shown in

Fig. 4.5(e). This distribution has greater sensitivity to the initial 2p configuration of the

decaying system [2]. In addition, the spin-singlet interaction in the p-p channel provides the

virtual state (“diproton”) which also can affect the long-range behavior of the correlations

(see [45, 46, 47] for the corresponding effects in 2n decay). The theoretical prediction for

ρcut = 105 fm in Fig. 4.5(f) reproduces the experimental data quite well, however, the

sensitivity to ρcut is diminished compared to the core-p energy distribution.

50

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

15000

1

Exp. + MC

(b) Exp.

cut (fm) 100 103

105

Even

ts

= Ecore-p /ET

105 103 100

(a)

theory"Y"

cut (fm) 60

dj/d

0.0 0.2 0.4 0.6 0.8 1.00

10

1

1.976 1.676

1.476theory

(d)

"Y"

ET (MeV) 1.276

dj/d

= Ecore-p /ET

10 1.23 0.7

1.45 0.9 1.45 0.7

theory dj/d

(c) Er (MeV) (MeV)

1.45 0.5

0

1

2

0.0 0.2 0.4 0.6 0.8 1.00

200

400

105 103 100

theory

(e) cut (fm)

60

"T"

dj/d

Exp. + MC

(f)Exp.

cut (fm) 100 103

105

Even

ts

= Epp /ET

Figure 4.5: Panels (a)-(d) show energy distributions in the Jacobi “Y” system where (a) givesthe sensitivity of the predictions to ρcut, (c) to the 15Fg.s. properties, and (d) to the decayenergy ET . Panels (e), (f) show energy distributions in the Jacobi “T” system where (e)gives the sensitivity to ρcut. The theoretical predictions, after the detector bias is includedvia the MC simulations, are compared to the experimental data in (b) and (f) for the “Y”and “T” systems respectively. The normalization of the theoretical curves is arbitrary, whilethe MC results are normalized to the integral of the data.

In the model the very long distances are achieved by classical extrapolation. This ap-

proximation has been studied using calculations with simplified Hamiltonians where it was

demonstrated that the classical extrapolation provides stable results if the starting distance

ρmax exceeds some hundreds of fermis for ET ∼ 1 MeV [41]. (e.g. ∼ 300 fm for 19Mgg.s.

decay where ET = 0.75 MeV). At such distances, the ratio of the Coulomb potential to the

kinetic energy of fragments is of the order 10−2–10−3. Figure 4.6 shows that for 16Neg.s.,

the predictions are consistent with the data only if the conversion from quantum to classical

dynamics is made at or above 200 fm.

4.6 Summary

The continuum of 16Ne has been studied both experimentally and theoretically with emphasis

on the ground state which decays by prompt two-proton emission. The measured decay

correlations in this work were found to require a theoretical treatment in which the three-

body Coulomb interaction is considered out to distances far beyond 103 fm. Our theoretical

treatment is now validated for use in interpreting the results of future studies of heavier

51

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

2000

Exp. + MC

"Y"

Exp.

max (fm) 100 200 1000

Even

ts

= Ecore-p /ET

Figure 4.6: The core-proton relative-energy distribution (“Y” system) obtained by classicalextrapolation started from different ρmax values.

two-proton decay with particular emphasis on extracting nuclear-structure information from

correlation observables.

We extract a limit of Γ < 80 keV for the intrinsic decay width of the ground state,

and while this is not inconsistant with some of the previous measurements, it is the first

measurement consistent with the theoretical predictions. All conclusions of this work were

only possible due to the high statistics and fidelity of the present measurements.

52

Chapter 5

16Ne Excited States

5.1 Background

Two-proton (2p) decay is most commonly considered as either of two extreme types: prompt

or sequential. In the first case, which is also called “true” 2p decay, the two protons are

emitted simultaneously in a three-body process. In sequential 2p decay, the two protons are

emitted in two distinct steps of binary decay. If the intermediate system formed after the

first step of single-proton decay is long-lived (narrow), then there will be no interactions

between the two protons. Ground-state true 2p emitters occur where either no intermediate

state is accessible via single-proton emission and/or the width of the intermediate state is

wide compared to the single-proton decay energy [2]. For the excited states of such nuclei,

these conditions may no longer apply and as the intermediate states become fully accessible

by single-proton decay, it has generally been assumed that sequential decay will prevail. In

this work we investigate such a case and show the transition from prompt to sequential is

nontrivial.

In the previous chapter, I concentrated on the ground-state of 16Ne which belongs to

the class of true two-proton emitters [28, 2] because the 15F s1/2 ground state (g.s.) is

not fully accessible by single-proton decay as indicated in the decay scheme of Fig. 5.1. The

53

experimental data used in that chapter has high statistics and excellent resolution allowing us

to isolate the effect of the long-range, three-body, Coulomb interactions on the momentum

distributions of the decay fragments. This effect was well reproduced by our three-body

model thus validating its use with heavier 2p decays and allowing one to separate out the

finer aspects of the decay dynamics.

Here, I continue the study of the 16Ne continuum and consider the decay of the first

excited, Jπ = 2+ state of 16Ne. From the level scheme in Fig. 5.1, one expects that the

decay of this state is sequential as the 15F s1/2 g.s. is fully accessible via one proton decay and

the decay energy for this step is large compared to the width of this state. In this case, the

decay energy could be efficiently shared between the degrees of freedom (first emitted proton

and the motion of 15F), which should provide favorable conditions for penetration. However,

both the experimental data and calculations indicate that the real situation is much more

complicated.

A similar problem was encountered in previous studies of the 6Be continuum in Refs.

[36, 38]. That work searched for the decay energy ET at which true 2p decay is replaced

by the sequential decay mechanism. In contrast to expectations, they found that such a

transition did not take place in the whole observed energy range (ET < 10 MeV). The decay

remained of a complicated three-body nature even if some aspects of the decay correlations

resembled the pattern expected for sequential decay. We now find similar results in the decay

of the first-excited state of 16Ne.

In 16Ne, the Coulomb interaction is ∼4 times stronger than in 6Be giving rise to higher

Coulomb barriers and thus narrower states. Another difference is that the valence protons

in 16Ne belong to the s-d shell and is built on positive-parity single-particle excitations. It

is interesting to see if these factors give rise to qualitative differences in the evolution of the

decay mechanism with decay energy. This study will help us to predict the behavior for

other s-d shell 2p emitters, for which similar quality data is not available.

Information on higher-lying excited states of 16Ne is very sparse, apart from the observa-

54

tion of another excited 2+ state in the 14O+p+p exit channel at ET ∼ 7.64 MeV [43, 44, 5]

and the possibility of a second 0+ state at ET ∼ 3.5 MeV [16]. At even higher excitation

energies, the 13N+p+p+p exit channel opens and from its measured invariant-mass spectrum

we find two new excited states.

5.2 14O+p+p Excitation spectrum

The distribution of 16Ne decay energy ET deduced from detected 14O+p+p events via the

invariant-mass method is displayed in Fig. 5.2. This spectrum has been corrected for a

contamination from 15O+p+p events where a small fraction of 15O fragments have leaked

into the 14O gate in the E − ∆E spectrum. This correction is described in more detail in

Chapter 4 where the raw uncorrected spectrum is also shown, Fig. 4.1.

The 16Ne ground-state peak at ET = 1.466(20) MeV dominates the spectrum, but in this

chapter I will concentrate on the smaller peak at ET = 3.16(2) MeV which has an excitation

energy of E∗ = 1.69(2) MeV. In the 16C mirror nucleus, the lowest excited states are a 2+1

state at E∗ = 1.766 MeV and a 0+2 state at E∗ = 3.027 MeV [62]. If the excited state

observed in this work is the 0+2 state, then this would correspond to a very large Thomas-

Ehrman shift of 1.3 MeV that would be ∼ 1 MeV larger than that for the ground state.

Recent theoretical calculations of the Thomas-Ehrman effect for 16Ne-16C mirror partners

[63] predicted shifts 16Ne 0+1 and 2+ states are around 200− 300 keV. Therefore we believe

this state at E∗ = 1.69 MeV is the mirror of the 2+1 first excited state of 16C. The energy

of this peak is close to other levels observed in previous experiments, see the discussion of

Sec. 5.7.

5.3 Width of the 2+ state

Figure 5.2 shows a fit to the 2+ peak using a Breit-Wigner line shape where the effects of the

experimental resolution (and its uncertainty) are included via Monte Carlo simulations. The

55

'ET

2+ 7.77

2+ 6.59 3 6.27 0+ 5.92 1 5.17

12.23

9.84

(2+) 7.64

4.62813 N p p p

Er

sequential?democratic / true 2p ?

2+ 3.16

16 Ne 15 F p 14O p p

0+ 1.466

5/2 2.80

1/2 1.23-1.52

true 2pET

Figure 5.1: Low-lying levels of 16Ne observed in this work and Ref. [5] and levels importantfor proton and multi-proton decay to 15F, 14O, and 13N states. The experimental uncertaintyof the 15F g.s. energy is indicated by hatching. Note a reduction 1.5 in the scale above the13N+p+p+p threshold. All level energies are given with respect to the 14O+p+p threshold.

56

Ne) [MeV]16 (TE1 2 3 4 5

Co

un

ts

0

200

400

600

+g.s. 0

+2B1 B2

Figure 5.2: Distribution of 16Ne decay energy determined from detected 14O+p+p eventsshowing peaks associated with the ground and first excited states of 16Ne. The solid curveshows a fit to the first excited state where the dashed curves indicates the fitted backgroundunder this peak. Two background gates, B1 and B2, are shown which are used to investigatethe background contributions to the measured decay correlations.

57

fit is shown by the solid line while the dashed curve indicates the fitted smooth, almost flat

background. This fit is not unique as different functional forms of the background could be

assumed. However, the relative magnitude of the background is small and the dependence

on the exact form of the background is not large (see below). The extracted intrinsic width

in this particular fit is Γ = 150(50) keV where the error bar results predominately from the

uncertainty associated with the effects of energy loss and small-angle scattering of the decay

products in the target. See Ref. [5] for a discussion for this uncertainty. Upper limits to the

width of this state were also obtained from the other neutron knockout experiments. Our

extracted width is consistent with Γ = 200(200) keV obtained in Ref. [43], but inconsistent

with the limit of Γ ≤ 50 keV from Ref. [44], see further discussion in the Section 5.7.

We have explored whether other choices of the background shape could result in signifi-

cantly smaller values of Γ. If the background has a minimum under the peak, the extracted

width would be smaller if the intrinsic width of this hypothetical minimum was less than

200 keV. Such a narrow “V”-shaped background seems unlikely, but in any case the depth of

the such a minimum is limited as the intrinsic background (before the effects of detector res-

olution is applied) must always be positive. With this constraint, the observed background,

after the effects of the experimental resolution is included, is always quite shallow for such

narrow “V” backgrounds and produces a negligible change in the extracted value of Γ. On

the other hand the use of wide “V”-shaped background can increase the extracted width.

We estimate the maximum value of Γ from this and other uncertainties is 250 keV and finally

we obtain Γ = 175(75) keV.

5.4 Theoretical models

We continue to exploit the theoretical models developed by L. Grigorenko et al. To study the

excited states we compare our data to two models both treating the decaying 16Ne system

as an inert 14O core interacting with two protons. The first model is semianalytical with a

58

simplified Hamiltonian, and the second is a rigorous three-body cluster model. In principle

three-body modles can yield: resonance energy, width as well as the correlations between

the momenta of the decay products. The rigorous three-body model provides predictions for

all of the above quantities while the simplified model is used to evaluate the width and, in

case of sequential decay, the correlations can also be described reasonably well.

5.4.1 Simplified decay model

The width and momentum distributions in the “Y” Jacobi system can be estimated within

the so-called “direct decay” R-matrix model presented in Ref. [2] where each proton is

assumed to be in a resonant state of the core+p subsystem with resonant energy Eji . The

differential flux for a system of total spin J is given by

dj(J)j1j2

(ET )

dε dθk=

ET ⟨V3⟩2

2πf(J)j1j2

(θk)

× Γj1(εET )

(εET − Ej1)2 + Γ2

j1(εET )/4

× Γj2((1− ε)ET )

((1− ε)ET − Ej2)2 + Γ2

j2((1− ε)ET )/4

, (5.1)

where ji is the angular momentum of a core+pi subsystem. In this double differential, one

variable is over the energy partition (ε) and the other the angular variable (θk), i.e. the Jacobi

variables. This model corresponds to the simplified Hamiltonian of the three-body system

in which the two protons interact with the core, but not with each other. It approximates

the true three-body decay mechanism in a fashion that provides a smooth transition to the

sequential-decay regime [59, 64]. It is essentially a single-particle approximation treating the

decays of configurations [j1j2]J independently and neglecting interactions between the valence

nucleons. The quantity Γji(E) is provided by the standard two-body R-matrix expression

for the decay width as function of energy for the core+pi resonance. The values Γji(Eji)

correspond to the empirical values of the resonance widths in both core-nucleon subsystems

i.

59

The angular-distribution function f(J)j1j2

(θk) in this model is obtained by coupling the

single-particle angular functions with momenta j1 and j2 to total momentum J . This function

does not always give realistic angular distributions [2]. However for the [sd] configurations,

dominating in the decay of the 16Ne Jπ = 2+ state, f(J)j1j2

(θk) is expected to be isotropic.

The matrix element ⟨V3⟩ can be well approximated by

⟨V3⟩2 = D3[(ET − Ej1 − Ej2)2 + (ΓJ)2/4] ,

where the parameter D3 ≈ 1.0− 1.5 (see Ref. [64] for details) and ΓJ , the three-body decay

width, is obtained selfconsistently from the FWHM of the distribution specified by Eq. (5.1).

The differential flux dj in Eq. (5.1) is normalized so that the three-body decay width is

obtained by integrating over the two Jacobi coordinates,

Γ(J)j1j2

(ET ) =

1∫0

dε

1(cigar)∫−1(diproton)

dθkdj

(J)j1j2

(ET )

dε dθk.

Typically for ground states there is only one configuration contributing significantly to the

width, however for the 2+ state, several configurations have non-negligible contributions and

the total width is given by

Γ(J)(ET ) =∑j1j2

W(J)j1j2

Γ(J)j1j2

(ET ) , (5.2)

where Wj1j2 are the weights of different configurations in the wavefunction (WF) for compo-

nents with p-core spins of j1 and J2.

5.4.2 Three-body calculations

The three-body 14O+p+p cluster model for the 16Ne decay was originally developed by

L. Grigorenko et al in Ref. [48]. Since that time this theoretical approach has developed

60

considerably [2] including a careful consideration of the Thomas-Ehrmann effect [63] and

a precise treatment of the decay dynamics and long-range Coulomb effects as described in

Chapter 4 and Ref [5]. The Thomas-Ehrmann effect is an energy-level shift seen for mirror

states where in one you have a neutron in a s1/2 orbital, and in the other it has been changed

to a proton. Since this orbital has no angular momentum barrier, the Coulomb interaction

will push the radius out to lower the energy. This is seen quite prominently in the first-

excited states of 13C and 13N where they energy for the 1/2+ level has shifted down in 13N

(which has a proton instead of a neutron) by ∼ 700 keV. Reference [63] provides a detailed

up-to-date description of the model used for these 16Ne calculations.

5.4.3 Three-body calculations

For the ground state of 16Ne, our three-body calculations predict Γ = 3.1 keV which was

found to be consistent with the upper limit we extracted for the intrinsic width of Γ < 80

keV [5]. Moreover, the model reproduces the experimental momentum correlations between

the decay products with high precision. The decay mechanism of this state is the so-called

true three-body decay and the hyperspherical method delivers very accurate results in this

case.

The 2+ state of 16Ne can decay sequentially via the 15F s1/2 g.s. at Er ≈ 1.4 MeV and

the first-excited, d5/2 state at Er = 2.80 MeV (see Fig. 5.1). The hyperspherical method is

not as well suited for width calculations of states with important sequential decay channels

and thus special care is required for the 2+ calculation. For this level, the three-body

calculations predict an intrinsic width of Γ = 51 keV for the largest basis size considered.

The basis convergence of this result is studied in Fig. 5.3. The basis size is defined by Kmax,

the maximum value of the principle quantum number K in the hyperspherical method. In

our calculations, large basis sizes become available with the help of an adiabatic procedure

(“Feshbach reduction”, see e.g. Ref. [59]) that converts the calculation to one with a smaller

basis size KFR, which is then treated in a fully dynamical manner. It can be seen from

61

10 20 30 40 50 60 7020

30

40

50

60

KFR = 16

(ke

V)

Kmax

(a)

4 8 12 16 20 24 28 3220

30

40

50

60

asy = 54.9 keV

(b)

calculation Kmax = 70

exp. fit upper exp. fit lower

KFR

asy = 55.8 keV

Figure 5.3: Convergence of the width calculations for the 16Ne 2+ state. (a)Kmax convergenceat a fixedKFR. (b)KFR convergence at a fixedKmax. Exponential extrapolations (separatelydone for odd and even KFR/2 values) point to a width of around Γ = 56 keV.

Fig. 5.3(a), that Kmax convergence for a given KFR is quite convincing. In contrast, KFR

convergence is not complete and displays some odd-even staggering with respect to KFR/2.

To estimate the final converged width at largeKFR, our collaborator L. Grigorenko performed

exponential extrapolations separately for both odd and even KFR/2 values [Fig. 5.3(b)]. The

extrapolations provided very similar asymptotic values of Γ ∼ 56 keV. Thus we regard this

as the correct theoretical prediction. This value is smaller than the experimental result

Γ = 175(75) keV, and we do not see how the three-body calculations could be improved to

match the experimental value.

5.4.4 Simplified decay model estimates

To investigate whether larger theoretical widths are possible, our collaborator performed

calculations using the simplified decay model of Eq. (5.1). For the estimates provided in

Table 5.1, he chose parameter sets in a reasonable way, by stretching all of them in the

direction maximizing the width estimate. The channel radius rch = 4.25 fm and the reduced

width θ2 = 1.5 were used for all core+p resonances. The energies of the p1/2 and p3/2

62

resonances were assigned assuming isobaric symmetry between 15F and 15C states. The

assumed energy of the d3/2 resonance Ed3/2 = 8 MeV is also the minimal value which does

not contradict the spectrum of 15C.

The structure information on the Jπ = 2+ 16Ne state in Table 5.1 is taken from Ref.

[63]. However, for this table we required a more detailed decomposition than was provided

in the that work. In spite of the simplicity of the model, the total width of the 2+ state

evaluated according to Eq. (5.2) is found to be around Γ = 51 keV in a good agreement with

the three-body calculations of Sec. 5.4.3 (I remind the reader that this value is inconsistent

with our experimental value).

As shown in Table 5.1, there are two ways to increase the estimated width: (i) decrease

in the energy Er of the 15F s1/2 ground state or (ii) drastically increase the [s1/2d5/2]2+

configuration weight. In both cases, the calculated widths remain at the lower limit of the

experimental uncertainty. In evaluating these possible modifications one should consider

their consistency with structure of the mirror states. The Thomas-Ehrman effect and the

Coulomb displacement energies for the 0+ and 2+ states of the 16Ne-16C isobaric mirror

partners were studied in detail in Ref. [63]. A sensitive relationship was found between the

energies of the 0+ and 2+ states, their structures, and the 15F s1/2 ground-state energy. The

latter was fixed in the calculations at Er = 1.405(20) MeV using our experimental values

for the ground and 2+ states of 16Ne, Chapter 4. It is very difficult to drastically change

the spin structure of predicted WF without a catastrophic effect on the consistency with the

experimental energies of the 0+ and 2+ states achieved in Ref. [63]. Thus, a complete revision

of the theoretical understanding of 16Ne and 15F systems is required to allow realization of

the variants proposed in (i) and (ii).

5.4.5 Outlook for 2+ state width

Given that the experimental width of the 2+ state is larger than expected, it is useful to

examine if there could be additional contributions to the 16Ne excitation spectrum which

63

Table 5.1: Simplified-decay-model estimates of important configurations of the 16Ne 2+ WFfor D3=1.0. The total decay width according to Eq. (5.2) is provided in the line “Total”.The sensitivity of the three-body width to variations of the 15F g.s. energy is illustrated bythe three bottom lines of the Table.

[lj1lj2 ]Jπ W(2)j1j2

(%) Ej1 (MeV) Ej2 (MeV) Γ(2)j1j2

(keV)

[p1/2p3/2]2+ 3.0 4.5 6.0 11[p23/2]2+ 1.1 6.0 6.0 7

[d23/2]2+ 1.2 8.0 8.0 0.04

[d25/2]2+ 5.4 2.8 2.8 1.6

[s1/2d3/2]2+ 71.3 1.4 8.0 50[s1/2d5/2]2+ 15.9 1.4 2.8 93

Total 97.9 50.9[s1/2d5/2]2+ 15.9 1.2 2.8 113[s1/2d5/2]2+ 15.9 1.3 2.8 103[s1/2d5/2]2+ 15.9 1.5 2.8 89

overlap with the 2+ state. One such possibility is that the peak we observed is actually

a doublet with contributions from both the 2+1 and the 0+2 levels. Fohl et al. [16] report

a 0+2 excited state at E∗ = 2.1(2) MeV [with corresponding ET = 3.57(20) MeV] in the

16O(π+,π−) reaction. This energy is a 2σ difference from our fitted centroid, but overlaps

with the observed peak.

In the three-body calculations [5], the 0+2 state is predicted in the range E∗ = 2.9 −

3.2 MeV, well separated from the observed peak and consistent with the energy of the E∗ =

3.03-MeV state in 16C which is expected to be the 0+2 mirror partner. Furthermore, the three-

body calculations of Ref. [5] indicate that the 0+ excitation function for 16Ne has a dip, and

not a peak, at the predicted energy of the 0+2 state, see Fig. 4.1. In these 17Ne(−n) →16Ne

neutron-knockout calculations , the 0+2 state is strongly suppressed as the overlap of the

valence-proton configurations for the 17Ne ground state with those of the 16Ne ground state

(0+1 state) is so strong that there is little room for yield from the 0+2 state. This assumed 17Ne

structure is regarded as quite realistic as it was found to lead to an excellent agreement with

a broad range of observables [49]. Of course, the 17Ne structure or the reaction mechanism

maybe more complicated than assumed in the calculations. Furthermore, the data of Fohl

64

-1.0

-0.5

0.0

0.5

1.0

exp"Y"

(a)

cos(

k)

0

10

20

30

40

50

60

(g)

p

p

14O

theo MC"Y"

(b)

0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.5

0.0

0.5

1.0

cos(

k)

exp "T"

(d)

0

10

20

30

40

500.0 0.2 0.4 0.6 0.8 1.0

theo MC "T"

(e)

0.0 0.2 0.4 0.6 0.8 1.0

simp MC "T"

(f)

simp MC"Y"

(c)

Figure 5.4: Comparison of the experimental Jacobi (a) “Y” and (d) “T” correlation dis-tributions to the those of the three-body model [(b) and (e)] and those from a sequentialdecay simulation [(c) and (f)]. The effects of the detector efficiency and resolution in thetheoretical distributions have been included via Monte Carlo simulations. (g) Shows therelative orientation and magnitude of the two proton velocity vectors for the peak regionsindicated by blue circles in panel (a).

et al. and the calculations of Ogawa et al. [65] suggest that the location of this state

could be much closer to our observed peak. In summary, it is clear that we do not fully

understand the magnitude of the width of the 2+ state and that this issue is strongly related

to excitation energy of the 0+2 state and its population in these neutron knockout reactions.

While the possibility of contamination from the 0+2 and width issue will remain unresolved,

the remained of this chapter will proceed assuming the state is entirely 2+ in character.

5.5 Correlations for 2+ state

The experimental two-dimensional Jacobi “Y” and “T” correlations for the 2+ state are

shown in Figs. 5.4(a) and 5.4(d), respectively. The ET gate used to select this peak is

indicated in Fig. 5.2. Both distributions show the presence of two ridge structures, and

65

indeed double-ridge structures are expected for sequential decay. The location of these

ridge structures along the Jacobi “Y” energy axis correspond qualitatively to those expected

from the decay via the s1/2 ground state of 15F with Er ∼ 1.4 MeV. However, these ridge

structures are very pronounced and intense only in the diproton halves of the distributions

[small ε in the Jacobi “T” or cos(θk) < 0 in the Jacobi “Y” distribution]. For the other

halves of the distributions, the overall intensity is reduced and the ridge structures have

largely faded out. These features can also be seen in the projected energy distributions

for the two Jacobi systems plotted in Figs. 5.5(a), 5.5(c), and 5.6. The geometry of the

most probable decay configurations, associated with the more intense regions of the ridges,

is illustrated in Fig. 5.4(g) (left).

5.5.1 Background contribution to correlations

Before discussing the correlations from our two models, it is useful to consider the contri-

butions of the background under the Jπ = 2+ peak. From the fit shown in Fig. 5.2, we

estimate that our energy gate spanning the excited-state peak contains a 24% background

contribution. Background gates, B1 and B2, were placed either side of this peak (see Fig.

5.2) to investigate the correlations associated with this background. The number of events

in these background gates is too small to obtain useful information from the two-dimensional

distributions, so we will concentrate on the projected distributions. The projected energy

distributions in both the “T” and “Y” systems are compared for the main and background

gates in Figs. 5.5(a) and 5.5(c). The results for the each background gates have been nor-

malized to 24% of the peak yield and the error bars are less than the size of the data points.

We should emphasize here that the “background” is expected to be largely a “physical

background” from decay of unresolved 16Ne states. For example, based on the theoretical

calculations of Ref. [5], the B1 gate contains contributions from the large high-energy tail of

the ground state and the B2 has contributions from a wide 1− state.

For the Jacobi “Y” system shown in Fig. 5.5(a), there are significant differences between

66

the two background distributions. The B2 distribution has a double-peak structure much

like that obtained for the excited-state gate, but the two peaks are separated further apart

in energy as would be expected for sequential decay of a higher-lying 16Ne level decaying

through the 15F ground state. In fact, the peak locations roughly match the expectation

from a sequential calculation with the simplified model (Sec. 5.5.2) which are indicated by

the arrows. For the B1 gate, there is only a single peak at ε ∼ 0.5, but this is expected

from Eq. (5.1) for a such small ET value. If the energy projection for the background under

the peak is intermediate between the B1 and B2 projections, then it will have a two-peak

shape similar to the peak-gated projection. Note the magnitude of the two peak structure

in the peak-gated distribution is too large to be entirely due to this background, therefore

the presence of the background only contributes moderately to this feature.

On the other hand for the Jacobi “T” system in Fig. 5.5(c), the two background dis-

tributions are quite similar and therefore it is reasonable to assume the background under

the peak has a similar, almost flat, dependence. As such, its contribution does not alter the

measured distribution significantly. The data in Fig. 5.5(d) shows the Jacobi “Y” energy

distribution after the average of the B1 and B2 gates are subtracted. It is not significantly

different from the original distribution in Fig. 5.5(c). A similar subtraction is also made for

the Jacobi “Y” distribution in Fig. 5.5(b), but in this case it is not clear how appropriate

this is. However, the change in shape from the original distribution in Fig. 5.5(a) is again

minor.

If the observed ET = 3.15 MeV peak is a doublet, as discussed in Sec. 5.4.5, and the

correlations for the two states are different, then one might see an ET dependence of the

correlations within the ET range of the peak. No evidence for such an effect was observed.

The only observed dependence of statistical significance is small shifting of the energies of

the two peaks which follows the expected ET dependence for decay through the 15F ground

state.

67

Cou

nts

0

200

400

600

800

Y

B2B1

(a) (b)

T)/E

core-p(Epp = Eε

0 0.2 0.4 0.6 0.8 1

Cou

nts

0

200

400

600T

(c)

T)/E

core-p(Epp = Eε

0 0.2 0.4 0.6 0.8 1

(d)

Figure 5.5: Comparison of projected Jacobi (a) “Y” and (c) “T” energy distributions forpeak-gated events and the events in the neighboring background gates B1 and B2. In (b)and (d) the experimental projections have been background subtracted (see text) and arecompared to the predictions of the three-body model before (dashed curves) and after (solidcurves) the effects of the detector bias and acceptance was included. The arrows in (a) showthe predicted locations of the maxima from Eq.(5.1) for the mean energies of each of thethree gates.

68

5.5.2 Correlations in the simplified model

In the simplified model [Eq. (5.1)], the double-ridge distribution of Fig. 5.4(a) is dependent

on the energy Er of the ground state s1/2 resonance in 15F. However, this energy is not well

defined experimentally with results varying from 1.23 to 1.52 MeV, see e.g. the discussion and

references in Refs. [60, 63]. Figure 5.6 demonstrates the strong sensitivity of the simplified

model to the value of Er. This result, for the first excited state, is in stark contrast to the

ground state where we found the “Y” system energy distribution was insensitive to realistic

variations of Er. The range of Er consistent with the data in Fig. 5.6 is 1.4 . Er . 1.5 MeV

which is also consistent with the range 1.39 . Er . 1.42 MeV determined from an analysis

of Coulomb displacement energies [63]. Also see the discussion of the 15F ground state in

Sec. 5.4.4.

A detailed view on the correlations from the simplified model is presented in the two-

dimensional distributions shown in Figs. 5.4(c) and 5.4(f). These distributions were calcu-

lated with Er = 1.405 MeV [63] and the effects of the detector efficiency and resolution were

included via the Monte Carlo simulations. Both distributions show the expected double-ridge

structures. The sequential aspect of the decay is most easily understood in the Jacobi “Y”

distribution of Figure 5.4(c). Here the two-ridge structures are prominent over the whole

angular range. The angular distribution functions f(J)j1j2

(ck) are isotropic for all the contribu-

tions in Table 5.1 except for the minor [p23/2]2+ and [d25/2]2+ components. Therefore ridges are

predicted to have uniform intensity as a function of cos(θk) and the small dependence seen

in Fig. 5.4(c) is just due to the detector acceptance. The use of smaller values of Er in these

simulations would result in the same basic picture, but the separation between the ridges

would be increased (as the two decay steps would become more skewed in Ecore−p partition)

and the match with the experimental ridges at low ε values would be worse.

Comparing the data and sequential simulations, we find that the experimental distribu-

tion looks sequential only in the regions were there are strong p-p final-state interactions

suggesting that the decay is sequential only when it is also of diproton nature and vice versa

69

T/Ecore-p = Eε0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cou

nts

0

100

200

300

400

500

600

700

800

900 [MeV]rE

1.05

1.20

1.35

1.50

1.65

Figure 5.6: Experimental energy distribution for 16Ne 2+ state in the “Y” Jacobi system(data points) is compared with distributions calculated by the R-matrix-type expression ofEq. (5.1) for the different s1/2 g.s. resonance energies Er in 15F listed in the figure. Thedetector efficiency and resolution were included via the Monte Carlo simulations.

70

which seems nonsensical. The “diproton” regions are those with small ε in “T” system and

cos(θk) ∼ −1 for medium ε values in “Y” system. The actual geometry of the most probable

decay configuration is visualized in Figure 5.4(g). Clearly the decay mechanism is more

complicated than this intuitively-simple sequential picture.

The apparent conflict between features that look sequential in the “Y” energy distribution

and features indicating strong p-p final-state interactions in the “T” energy distribution was

also noted in the decay of highly-excited states of 6Be [36]. As the 6Be results are qualitatively

consistent with the present work, then it seems that the detailed structure of the state and

the magnitude of the Coulomb barriers are not responsible for explaining these common and

unexpected correlations.

5.5.3 Comparison with three-body calculations

The predicted two-dimensional correlations from the three-body model, after the effects of

detector bias and resolution are included via the Monte Carlo simulations, are shown in

Figs. 5.4(b) and 5.4(e) for the two Jacobi systems. These three-body calculations reproduce

the major features of the experimental results. Like the experimental data, the “sequential-

decay” ridges are clearly present and pronounced only in the diproton region. The agreement

is not perfect. For example, the three-body calculations predict a “hole” in two-dimensional

distributions that is most obvious in the Jacobi “Y” distribution of Fig. 5.4(b) where it is

centered at ε = 0.5, cos(θk) ∼ 0.3. This feature, if present at all in the experiment data,

is significantly reduced. However, it should be noted that the experimental distribution are

expected to contain ∼ 24% background which will make such a fine feature difficult to see.

More detailed quantitative comparisons are made in Figs. 5.5(b) and 5.5(d), where the

projected energy distributions are compared to the data. The predictions before and after

accounting for the effects of the detector bias and resolution are similar as indicated by

the dashed and solid curves, respectively. The two-peak structure is well reproduced in the

projection for the Jacobi “Y” distribution in Fig. 5.5(b). In the Jacobi “T” distribution of

71

Fig. 5.5(d), the three-body model also predicts an appropriate enhancement of the diproton

region (small ε), but the magnitude of the effect is somewhat larger than that observed

experimentally. As mentioned above, a similar result was obtained for the highly-excited

continuum of 6Be [36]. We note that computation of the “Y” ε distribution is sensitive

to model parameters like the resonance energy Er of 15F ground state. In contrast, the

“T” ε distribution is much less sensitive to these parameters and very stable in various

computation conditions. It is also possible that this disagreement is related to uncertainties in

the background contribution, however it is the “T” background that seems better constrained

than the “Y” background (Sec. 5.5.1).

5.6 Decay mechanism for 2+ state

The three-body calculations reproduce the diproton and sequential features of the experi-

mental distributions. In order to better understand the origin of these unusual momentum

correlations we have examined the radial evolution of the decay WF Ψ(+). Figure 5.7 shows

the correlation density

W (ρ, θρ) =

∫dΩx dΩy |Ψ(+)(ρ, θρ,Ωx,Ωy)|2

as a function of the hyperangle and hyperradius in the Jacobi “Y” system. The Jacobi

vectors X and Y in this system, illustrated in Fig. 5.7(c), are defined via the hyperspherical

variables as

ρ2 = A1A2

A1+A2X2 + (A1+A2)A3

A1+A2+A3Y 2 = 14

15X2 + 15

16Y 2 ,

X = ρ√

A1+A2

A1A2sin(θρ) = ρ

√1514

sin(θρ) , (5.3)

Y = ρ√

A1+A2+A3

(A1+A2)A3cos(θρ) = ρ

√1615

cos(θρ) , (5.4)

where in the “Y” system A1 = Acore and A2 = A3 = 1. The mid point of the θρ -axis in Figs.

72

020

4060

800.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

3025

20

2.5 5 5

2.5

2025

(iv)

(iii)

(i)

(fm

)

(rad)

30

(ii)

(a)

020

040

060

080

0

(b)

(fm

)

1.1E

-22.9E

-27.2E

-21.8E

-14.5E

-11.1E

02.9E

07.2E

01.8E

14.5E

11.1E

2Ja

cobi

"Y"

x 10

x 10

x 10

x 10

x 10

x 10

x 10

x 10

x 10

x 10

x 10

15F(c.m.)

Y

(c)

p 2

p 1

14OX

Figure

5.7:

Correlation

density

W(ρ,θ

ρ)for

16Ne2+

stateW

FΨ

(+)in

the“Y

”Jacob

isystem

.Pan

els(a)an

d(b)show

the

evolution

ofthepredictedhyperan

gulardistribution

withhyperradiuson

twodifferentscales.

Yellow

curves

in(a)provide

thelevels

withconstan

tab

solute

values

ofX

andY

vectorsindicated

bytheattached

numbers(infm

).Blue(dashed)an

dpink(solid)arrowsqualitativelyindicatetheclassicalpathsconnectedwithsequential

andthe“tethered”decay

mechan

isms

correspon

dingly.

Thebluefram

ein

thepan

el(b)show

sthescaleof

pan

el(a).

Pan

el(c)illustratesthearrangementof

theX

andY

vectorsforthe“Y

”Jacob

isystem

,seeEqs.

(5.3)an

d(5.4).

73

5.7(a,b) approximately corresponds to equal projections of the hyperradius on the X and Y

vectors defining the positions of p1 and p2 respectively, i.e.

θρ ∼ π/4 → X ∼ Y ∼ ρ/√2 .

For large ρ values, the coordinate-space hyperangle θρ transforms into momentum-space

hyperangle θκ defining the energy distribution between the subsystems:

θρ → θκ, Ex = ET sin2(θκ), Ey = ET cos2(θκ). (5.5)

Thus the representation of Figs. 5.7(a,b) best illustrates the geometry of the single-particle

configurations.

The population density in Fig. 5.7(a) is very large for typical nuclear dimensions ρ < 5 fm

where the two protons are inside the Coulomb barrier. This indicates the region of resonant-

state formation. Extending out from this region are ridges associated with different decay

paths.

Classical trajectories for sequential decay, differing in the decay time of the 15F interme-

diate state are indicated by the dashed (blue) arrows. The initial portion of these trajectories

is common to all sequential decays (one of the protons is emitted while the other remains

close to the core) and they follow lines of roughly constant, but small, values of either X or

Y . The presence of the ridges in Fig. 5.7(a), which are roughly parallel to the yellow lines

shown for constant X = 2.5 or Y = 2.5 fm, confirms that there is a sequential component in

these three-body calculations. The classical trajectories deviate from these ridges when the

second proton is emitted and at this stage the classical trajectories diverge and so this part

of the decay is not reflected by a ridge structure in the WF and thus is less visible. However,

the decay paths do eventually concentrate at larger ρ values along the two well-defined ridges

which are clearly seen in Fig. 5.7(b).

The other prominent penetration path in the three-body calculations, schematically in-

74

dicated by solid (pink) arrows, is more unusual. One observes a well-defined ridge in the

range (i)-(ii) in Fig. 5.7(a) extending out from the main concentration of contours. For this

ridge, X ∼ Y and thus the protons are penetrating the barrier simultaneously. The density

in this ridge within the range (i)-(ii) again decreases exponentially with ρ. Simultaneous

emission of protons is of course a well established decay mechanism [2]. However, the emis-

sion dynamics changes in the range (ii)-(iii). Here the arrows more closely follow the lines of

constant X and Y which means that radial motion of one of the protons is now practically

stopped (much slower than the orbital motion), while the other continues to move in the

radial direction. Finally in the range (iii)-(iv) are two symmetric ridges which smoothly

evolve with ρ to the “sequential-decay” double-hump structures observed in the energy dis-

tribution of Fig. 5.5(b). This region is common for both the initial sequential and prompt

decay paths and both protons are more or less in the state of free flight except for the effect

of the long-range Coulomb interaction. This behavior implies that in the evolution along the

path (i)-(iv), the “near” proton leaves its position adjacent to the core at point (iii) starting

the expansion stage of the whole system. At this moment even the “near” proton is already

far beyond the typical nuclear size relative to the core and thus the WF component evolving

along the (i)-(iv) pathway cannot “remember” information about core-p interaction. How-

ever, the sequential trajectories merge with this decay path at this point and from Eq. (5.5),

the “near” proton and core have the relative energy appropriate for the 15F ground-state

resonance. Thus we conclude that the expansion stage (iii)-(iv) is initiated, for this WF

component, by the interference with WF components associated with the sequential decay

path [dashed (blue) arrows].

The angular distributions of the X and Y vectors is integrated for presentation of Fig.

5.7. The analysis of the WF for the “pink trajectory” region indicates that this angular

distribution is quite broad with a mean angle between X and Y close to π/2. The average

case of the solid (pink) path from Fig. 5.7 is visualized in Fig. 5.8. In the three-body decay

plane (rz = 0), the initial trajectories of protons are directed along the rx and ry axes. The

75

line rx = ry corresponds to the trajectory of the center of mass of the p-p subsystem. It

can be seen that at distances of about 30 fm, the radial propagation of one of the protons

is practically stopped and the protons behave for some time as connected with a kind of

“tether” of (practically) fixed length (blue dotted lines in Fig. 5.8) producing complicated

spatial trajectories. The realistic motion in our quantum-mechanical calculations does not

of course correspond to this piecewise trajectory; quantum mechanical motion is always

“smooth” and there are no well defined decay paths as in the classical case. However, this

idealized decay path is quite instructive and demonstrates the complexity of this decay that

we portray in qualitative terms as a “tethered decay mechanism”.

5.7 Previous Experimental Studies

In 1978 KeKelis et al. [17] reported the detection of the first excited state of 16Ne from the

observation of a peak with yield of ∼ 12 counts at ET = 3.03(7) MeV in the 20Ne(4He,8He)

reaction.

More recently, information on the lowest 16Ne excitations was obtained in two studies

[43, 44] which also used neutron knockout from 17Ne beams, but at relativistic energies. See

Table 5.2 for a comparison of extracted centroids and widths for the ground and first-excited

states in these and the present experiment.

The work of [43] uses a technique based on the tracking of the reaction products, a

technique created for studies of the radioactive decay lifetimes in the fs-ns range. In the

case of the much shorter-lived 16Ne states, the lifetime information is not extractable. The

spectroscopic properties are obtained from kinematically incomplete information. Therefore

the resonance parameters are recovered by MC simulations based on theoretical assumptions

about spectrum populations and decay mechanisms, both of which have uncertainties. The

16Ne g.s. position ET (0+) = 1.35(8) MeV found in [43] has a value that is somewhat lower

than ours, but otherwise the results are consistent.

76

The 16Ne excitation spectrum measured in Refs. [44, 66] is very similar to ours. This

is evidence for a similar reaction mechanism despite the large differences in bombarding

energies: E/A = 500 MeV compared to E/A = 57.6 MeV in our work. The experimental

resolution and statistical uncertainty in Refs. [44, 66] are significantly worse than in our work

[5], the former can be gauged by the widths of the experimental peaks that are almost a

factor of two larger than those in Figure 5.2. Despite these differences, it is surprising that

the uncertainties on the extracted widths from Refs. [44, 66] are smaller than ours. Such an

extraction would require an extremely precise understanding of their experimental resolution.

The 2+ decay energy ET in Refs. [44, 66] is consistent within the listed experimental errors

to our value, but the extracted width in these references (Γ < 50 keV) is inconsistent with

the present work.

Correlation data for the 2+ state in the relativistic study were also presented in Ref. [66]

and it was concluded that they are consistent with a purely sequential calculation. No

two-dimensional correlation plots are shown in Ref. [66], however, projections on the energy

and angular axes are presented (see Fig. 8 of [66]). Their energy distribution in the “Y”

system does not have the doubled-humped structure present in the distribution of Fig. 5.5(a).

However, if we take our best-fit simplified-model calculation or the three-body model result

and artificially decrease the experimental resolution in our MC simulations so as to reproduce

the experimental width of the 2+ state in the relativistic study, then this feature is washed

out and one is left with a broad peak similar to that found in Ref. [66].

The energy distribution in the “T” system obtained from the study of Ref. [66] is consis-

tent with their flat sequential calculation within their large statistical error bars. However,

it is also clear that a dependence with enhanced correlations in the diproton region of sim-

ilar magnitude to that found in our distribution (see Fig. 5.5(b)) would also be consistent.

Therefore, the two sets of experimental data may not be inconsistent, however more precise

comparisons would require a detailed knowledge of the experimental acceptance of Ref. [66].

It seems that the higher resolution and statistics of the our work have permitted us to bet-

77

ter characterize the decay mechanism and allow the non-sequential aspects of the decay to

become clearer.

5.8 Higher Excited States

In Ref. [5] we also reported on the a second excited state in the 14O+p+p channel at ET =

7.60(4) MeV. This peak was relatively weak in our excitation spectrum (due to its lower

detection efficiency) so no further analysis was attempted. Even higher 16Ne excited states

were observed in Ref. [43] by their feeding into the states of 15F. However, it was not possible

to resolve such states in this experiment.

At these higher excitation energies the decay will also start populating the 13N+p+p+p

exit channel (corresponding energy above its threshold is denoted as E ′T , see Fig. 5.1).

The decay-energy spectrum for this channel is shown in Fig. 5.9 and displays two peaks at

E ′T = 5.21(10) and 7.60(20) MeV corresponding to 16Ne excitation energies of E∗ = 8.37(10)

and 10.76(20) MeV, respectively. The solid curve shows the best fit for one choice of the

background contribution (dashed curve). As for the 14O+p+p channel, the fit assumes intrin-

sic Breit-Wigner line shapes where the effect of the experimental resolution is incorporated

via the Monte Carlo simulations. The background is relatively large and other functional

forms can give good fits as well. We have included contributions to the uncertainties of the

peak energies based on fits with other background choices. The fitted intrinsic widths of

Table 5.2: The properties of 0+ g.s. and first 2+ states of 16Ne obtained in the recentneutron knockout reaction studies with 17Ne beam. Energies and widths are in MeV. Allthe ET values are provided relative to the 14O+p+p threshold.

Work [43] [44] [5], this workJπ ET Γ ET Γ ET Γ0+ 1.35(8) 1.388(15) 0.082(15) 1.466(20) < 0.082+ 3.2(2) 0.2(2) 3.220(46) < 0.05 3.160(20) 0.150(50)(2+) 7.6(2) 0.8+0.8

−0.4 7.57(6) ≤ 0.1 7.60(4) ≤ 0.5(?) 9.84(10) 0.32(10)(?) 12.23(20) 0.51(23)

78

0 20 40 60 800

20

40

60

80

(iv)

(iii)

(ii)

(ii)

(i)

X

Y

= 65 fm

= 90 fm

= 65 fm

asymptotic

trajectory 1

pp c

ms

trajec

tory

r y (

fm)

rx (fm)

asym

ptotic

tra

jector

y 2

= 41 fm

= 90 fm

= 41 fm

(iii)

(iv)

Figure 5.8: Schematic of the tethered decay mechanism. Trajectory of two protons corre-sponding to solid (pink) path in Fig. 5.7. The configurations labeled by the Roman numeralscorrespond to the same configurations denoted in Fig. 5.7. The p-p center-of-mass motionpath rx = ry is collinear with Y vector for the Jacobi “T” coordinate system.

79

Ne) [MeV]16 (TE’4 6 8 10

Co

un

ts

0

50

100

150

200

Figure 5.9: Distribution of 16Ne decay energy determined from detected 13N+p+p+p events.Gates are shown on the two observed peaks which are used in the analysis shown in Figure5.10.

these peaks were found be Γ = 320(100) keV and 510(230), respectively.

With three protons in the exit channel, it is much more difficult to determine the decay

mechanism compared to channels with just two protons, especially given the relative large

background contribution. However some information on the decay path can be established.

Figure 5.10 shows the 14O excitation reconstructed from the core and one of the detected

protons for each of the two peaks in Fig. 5.9 using the gates shown by the dashed vertical

lines. The location of the first five excited states in 14O are shown by the vertical dashed

lines in this figure. In interpreting these spectra, one must remember that more than half of

the events come from background below the peaks. Also as only one of the three detected

80

protons can come from the decay of an excited 14O intermediate state, then at least 2/3 of

the remaining yield in these spectra is an additional background. Although we cannot rule

out small contribution from each of these 14O states in Fig. 5.10, a peak associated with the

forth excited state at E∗ = 6.59 (Jπ = 2+) is quite prominent for both 16Ne gates. Of course

it is not entirely clear whether this 14O level is associated with the background or the peaks

in the 16Ne spectrum. However, we note that gating on this Jπ = 2+ 14O peak, strongly

enhances the E ′T = 5.21 MeV 16Ne peak and thus we feel confident that this 2+ level is an

important intermediate state in the decay of this level. A similar search for possible 15F

intermediate states was fruitless probably because such states are expected to be wide at the

available 15F excitation energies in these decays.

5.9 Summary

The decay of the first excited state of 16Ne has been studied. A peak located at decay

energy ET = 3.16 MeV was observed in the 14O+p+p invariant mass spectrum which is

consistent with the expected location of the 2+1 first excited state based on the better known

spectroscopy of the mirror nucleus 16C. The intrinsic width of this state was determined to

be between 100 and 250 keV.

Three-body calculations which reproduced the momentum correlations of the 14O+p+p

decay products of the ground state and were consistent with the experimental limits for its

intrinsic width were found to predict an intrinsic width for the 2+1 state of Γ ∼ 56 keV which

is smaller than the experimental range. This conflict suggests either some deficiency in the

calculations or the possibility that the observed peak has other contributions, for example,

from the second 0+2 , state.

We have provided a detailed analysis of the dynamics of the three-body decay model that

has met with considerable success in reproducing many features of our and other group’s

data. The predicted 16Ne2+2p decay mechanism differs drastically from the common ideas

81

O) [MeV]14E* (5 6 7 8

Cou

nts

0

20

40

60

80

100

120

140

160 - = 1πJ+ = 0πJ - = 3πJ+ = 2πJ+ = 2πJ

(a)

O) [MeV]14E* (5 6 7 8

Cou

nts

0

20

40

60

80

100

120

140

160 (b)

Figure 5.10: Distribution of 14O decay energy determined for all possible 13N+p subsets ofeach detected 13N+p+p+p event. Pannel (a) is for the gate on the E ′

T = 5.21 MeV state in16Ne while (b) is for the E ′

T = 7.60 MeV state. The vertical dashed lines show the locationof the first five excited states in 14O.

82

about the decay process in terms of sequential or/and diproton decay mechanisms. The

two-dimensional correlations measured for the 2p decay of the first excited state shows a

double-ridge pattern expected for sequential decay only when the two protons have low

relative energy (diproton-like correlation). A careful investigation of the decay mechanism

on the level of the wavefunction in the three-body model indicates that the real situation is

quite intriguing. The model predicts both sequential and prompt-decay paths which interfere

and produce the strange decay pattern which we qualitatively describe as a “tethered decay

mechanism”. The observed correlation pattern is qualitatively similar to that measured

previously for excited 6Be fragments and suggests other examples will be found in the future.

Finally, new 16Ne excited states with total decay energy ofE ′T = 5.21(10) and 7.60(20) MeV

[corresponding E∗ = 8.37(10) and 10.76(20) MeV] were observed in the 13N+p+p+p exit

channel. There is some evidence that both these peaks have a sequential component in their

decay dynamics with intermediate populations of the E∗ = 6.59 MeV, Jπ = 2+ resonance in

14O.

83

Chapter 6

Isobaric Analog State in 16F

6.1 Background

In Chapter 3 evidence for a new type of direct 2p decay was presented, namely decay from

an Isobaric Analog State (IAS) to an Isobaric Analog State. The daughter nucleus then de-

excites to its ground state via γ-ray emission. While 8BIAS was the first case where all of the

decay products were measured (two protons, the 6Li core and the γ), it is not the only case

where this has been proposed as the decay mechanism. In some previous work by our group

a resonance was seen in the 2p+10B decay channel corresponding to a state in 12N [23]. If

the detected 10B were formed in its ground state, then the measured excitation energy would

not correspond to any known state in either 12N or its mirror 12B. However, 12NIAS should

decay to 10BIAS, in an analogous way to the decay of 8BIAS.10BIAS is known to decay via γ

emission, however the γ rays were not measured in that experiment. If this measured state

were the 12NIAS, and it decayed to 10BIAS, then the measured excitation energy of 12N is too

low by the excitation energy of 10BIAS. Adding in this missing energy brings the measured

energy up to the predicted excitation energy of 12NIAS from the IMME (see Chapter 7 for

more details on the IMME). Thus while this decay mode was not confirmed in this case, the

measurement of the decay mode 8BIAS lends strong support to this assumption.

84

ZT2− 1− 0 1 2

) [k

eV]

2 Z+c

TZ

M -

(a+

bT∆

40−

30−

20−

10−

0

10

20

30

40

Ne16 O16 N16 C16

Figure 6.1: Deviation from the fitted quadratic form of the IMME for the lowest T = 2states in the A = 16 isobar.

85

MeV

0

2

4

6

8

10

12

14

F16

, T=1-0

IAS, T=2+0

O15p+, T=1/2-1/2

+1/2

+1/2

+1/2

IAS, T=3/2+1/2

N142p+, T=0+1

IAS, T=1+0

n+15F, T=3/2+1/2

N12+α, T=1 +1

N13He+3

, T=1/2-1/2

, T=1-0

-1

-2

-3

O15p+

Figure 6.2: Partial level scheme for the decay of 16FIAS (level in green). Colored levelsindicate isospin allowed decays from the 16FIAS. All energies are relative to the ground stateof 16F. The inset shows an expanded level scheme for the low-lying states of 16F.

86

Moving higher in mass, the next Isobaric Analog State that could decay in this manner

is 16FIAS. While this resonance has never been observed, the energy can be predicted from

the IMME for A = 16, Fig. 6.1. The quadratic fit of the A = 16, T = 2 quintet contains

the new mass for 16Neg.s. from Chapter 4 and takes the other masses from the most recent

mass evaluation [67]. This fit predicts an excitation energy for 16FIAS of 10.12 MeV, which

is displayed as the green level in the partial level scheme, Fig. 6.2. As was true in both

8BIAS and 12NIAS, the only isospin- and energy-allowed particle decay is a 2p decay to the

isobaric analog state of the daughter, 14N. The 14NIAS is known to decay by emission of a

2.313 MeV γ ray to the ground state. If this 2p decay were to be observed in an invariant

mass spectrum, the peak would be located at ∼ 7.8 MeV.

6.2 Two-proton decay

With a 17Ne beam, 16F states can be formed in proton-knockout reactions. In the same

data set obtained to detect 16Ne two-proton decay, two-proton decay of 16F states were

investigated. The reconstructed excitation energy spectrum of 16F from all detected 2p+14N

events is shown in Fig. 6.3 (a). If 16FIAS were to decay by the 2p+14NIAS path, there should

be a peak at the position of the red arrow. An expanded version of this can be seen in the

inset to Fig. 6.3 (a). The observed peak is ∼ 200 keV lower than the expected position of

the IAS. While this does not rule out the possibility that it is the IAS, it is highly unusual

for the energy of an analog state to deviate that far from the quadratic form of the IMME.

If this were the IAS, it must be in coincidence with the 2.313 MeV γ ray from the decay

of 14NIAS. The energy spectrum of γ rays in coincidence with events in the region of the

observed peak are shown in Fig. 6.3 (b). For a 2.3 MeV γ ray, the photopeak efficiency of

CAESAR is ∼ 20%, i.e. with roughly 100 events in the peak we should measure 20 events in

the photopeak. We can therefore conclude that the observed peak does not decay through

the 14NIAS and is a T = 1 state at E∗ = 7.67 ± 0.02 MeV. This is likely the analog of the

87

E∗ = 7.674 MeV state in the mirror, 16N, for which no Jπ assignment was made.

As we did not see the 16FIAS in the isospin allowed 2p decay channel, we should look for

an explanation. We saw isospin allowed 2p decay from the isobaric analog states in both 8B

and 12N and that channel is also energetically allowed for 16FIAS. The decay energies for

the IAS to IAS transitions are listed in Table 6.1. As one moves higher in mass the decay

energy becomes smaller, and at the same time the Coulomb barrier becomes larger. Both of

these effects dramatically reduce the barrier penetration factors, Pℓ, for these decays. The

isospin allowed 2p decay for 16FIAS is more than 100 times slower than the isospin allowed

2p decay for 8BIAS. This increase in the partial half-life, decrease in partial decay width,

allows isospin non-conserving decay channels to be competitive.

Decay channel ET (MeV) ℓ Pℓ8BIAS → 2p+6LiIAS 1.312 0 0.090712NIAS → 2p+10BIAS 1.165 0 0.014416FIAS → 2p+14NIAS 1.046 0 0.00073616FIAS → 2p+14Ng.s. 3.359 0 0.28916FIAS → α+12Ng.s. 1.037 2 0.000034

16FIAS → 3He+13Ng.s. 0.523 1 7.0x10−8

Table 6.1: Decay energies (ET ) and barrier penetration factors (Pℓ) for the decay channelsrelevant to this work. The barrier penetration factors for 2p decays are calculated assumingthat each proton takes away 1/2 ET . Isospin non-conserving decays are listed in the lowersection of the table.

6.3 Isospin non-conserving decays

From phase-space considerations, one of the most competitive isospin non-conserving decay

mode for 16FIAS is 2p emission to the ground state of 14N either directly by simultaneous

2p emission or sequentially through an excited state of 15O. If the energy predicted from the

IMME is correct, then either possibility would produce a peak at the location of the black

arrow, Fig. 6.3 (a). While there is a peak near that energy in the 2p+14N invariant mass

spectrum, the excitation energy is E∗ = 10.26 ± 0.02 MeV, which is more than 100 keV

88

E* [MeV]4 6 8 10 12

Co

un

ts

0

200

400

600

800

1000

1200

1400

1600

E* [MeV]6 7 8 9

Co

un

ts

0

20

40

60

80

100

120

140

160

X10

(a)

[MeV]γE0 1 2 3

Co

un

ts

0

2

4

6

8

10

12(b)

Figure 6.3: (a) Excitation energy spectrum of 16F from all detected 2p+14N events. Theinset is shows the same histogram expanded in the region around the expected 16FIAS peak.(b) γ ray energy spectrum measured in coincidence with events inside of the gate indicatedin (a) with the blue dashed lines. The excepted photopeak position (2.313 MeV) is markedwith the red dashed line.

89

different than the predicted energy. This is also larger than any known deviation from the

quadratic IMME for an isospin multiplet. Therefore, without further evidence that this is a

T = 2 state, we must conclude that it is probably a previously unknown T = 1 state.

Other possible energy-allowed but isospin-forbidden transitions are alpha emission and

3He emission. Both decays have decay energies at or below 1 MeV and have angular momen-

tum barriers. The barrier penetration factors are much lower than even the isospin allowed

2p decay channel, so we do not expect to see any yield to these channels. Figure 6.4 shows

the excitation energy spectra for 16F decay to α (a) and 3He (b). No peak is observed near

10.12 MeV in either spectrum, indicating that 16FIAS does not decay through either of these

channels.

The final possibility for isospin non-conserving decay is via 1p emission. The 16FIAS has

Jπ = 0+, so it will preferentially decay to 1/2+ states. All energy allowed 1/2+ states in 15O

are shown in Fig. 6.2. The 1/2+ isobaric analog state in 15O (in red) is energy forbidden.

The next two highest 1/2+ states will decay to 14Ng.s. which was previously ruled out. The

last 1/2+ state, the first-excited state, is known to decay via a 5.183 MeV γ ray. The

excitation energy spectrum from all p+15O events is shown in Fig. 6.5. If 16FIAS decayed

through the first-excited state in 15O, this would be detected as a peak at ∼ 4.9 MeV in the

invariant-mass spectrum [E∗(16FIAS) - Eγ], and no peak is observed. While 1p decay to the

ground state is ℓ = 1, the decay energy is 10.655 MeV so the barrier penetration factor is

large. While no peak is observed at 10.12 MeV, we cannot rule out this possibility as our

detection efficiency is very small in this region.

6.4 Gamma decay

The only other energy-allowed, isospin-conserving decay mode for 16FIAS is γ decay to a low

lying state in 16F which will then 1p decay to 15Og.s.. The low-lying structure of 16F can be

seen in the inset of Fig. 6.2. While γ decay is not normally competitive with proton decay,

90

E* [MeV]9 9.5 10 10.5 11 11.5 12

Co

un

ts

0

50

100

150

200

250

300

N12 + α → IASF16

(a)

E* [MeV]9 9.5 10 10.5 11 11.5 12

Co

un

ts

0

20

40

60

80

100

120

140

160

N13He + 3 → IASF16

(b)

Figure 6.4: Excitation energy spectra for all detected (a) α+12N events and (b) 3He+13Nevents.

91

E* [MeV]0 2 4 6 8 10 12

Co

un

ts

1

10

210

310

410

Figure 6.5: Excitation energy spectra for all detected p+15O events. The blue arrow indicatedthe energy of the T = 1 state from the blue path, see text, if the γ-ray energy were added.

for this ∼ 10 MeV E1 transition (0+ → 1−) the estimated [Weiskopf] γ decay lifetime is

2× 10−18 s. As we do not see the isospin allowed 2p decay, the only energy-allowed particle

decays are isospin forbidden. This may make the particle decay lifetime long enough to be

comparable to that expected for this γ decay.

For a 10 MeV γ ray in CsI(Na), the attenuation coefficient (related to the interaction

probability) for pair production is more than a factor of three larger than Compton scattering,

with photoelectric effect many orders of magnitude less than both. In pair production, the

γ ray spontaneously creates a positron-electron pair that deposit all but 1.022 MeV of the

energy of the γ ray in the first crystal. That 1.022 MeV is released as two 511 keV γ

rays when the positron annihilates with an electron in the crystal. Those γ rays can be

detected in neighboring crystals and added to the energy measured in the first. Similarly,

γ rays which Compton scatter in the first crystal will deposit most of their energy in that

crystal. The remaining energy can be detected if the scattered γ is detected in a neighboring

crystal. Aside from small-angle Compton scattering, in both of these cases these γ rays will

over-range our ADCs.

92

[MeV]γE0 1 2 3 4 5 6 7 8 9 10

Co

un

ts

1

10

210 G1

(a)

E* [MeV]0.5− 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Co

un

ts

0

5000

10000

15000

20000

25000

30000

x20

(b)

Figure 6.6: (a) γ energy spectrum in coincidence with all p+15O events. (b) 16F excitationenergy spectrum of p+15O events. This is the same spectrum shown in Fig. 6.5 but witha linear ordinate. The blue dashed line marks the position of the first-excited state in 16F.The red dashed histogram is the excitation energy spectrum gated on the gamma rays ingate G1.

93

MeV

0

2

4

6

8

10

12

F16

, T=1- = 0πJ

-1

?

IAS, T=2+0

O15p+, T=1/2- = 1/2πJ

+1/2

Figure 6.7: Partial level scheme of 16F . The red arrows indicate a possible isospin-conservingdecay mode for the decay of 16FIAS, and the blue mode indicates the possible decay modeof a T = 1 state of 16F.

94

The γ-ray energy spectrum measured in coincidence with all detected p+15O events is

shown in Fig. 6.6 (a). While we would expect a smoothly falling distribution that cuts

off around 8.5 MeV, the maximum energy that does not over range our detectors, we see

a wide peak centered at ∼ 5 MeV. Gating the p+15O excitation energy spectrum on this

peak pulls out predominantly a single peak, Fig. 6.6 (b). Two possible explanations for this

particle-gamma coincidence are summarized in Fig. 6.7. If the γ decay preceded the particle

decay, the invariant-mass peak corresponds to the 0.193 MeV first-excited 1− state in 16F

(red path). If the 16FIAS were to γ decay, it would preferentially decay to this first-excited

state. However, there is no obvious reason for a 10 MeV γ ray to form a peak at ∼ 5 MeV.

Another possibility is that the particle decay precedes the γ decay (blue path). If that were

the case, then the particle decay is to the first-excited state of 15O, which γ decays via a

5.183 MeV γ. Then this particle + γ decay would correspond to the decay of a state at ∼

5.4 MeV in 16F that is T = 1. The position of this low T state is marked by a blue arrow in

Fig. 6.5. While the blue decay scheme is possible, there would need to be a nuclear structure

reason for that state to decay to the excited state of 15O, giving up a large amount of phase

space, rather than decaying directly to the ground state.

6.5 Summary

We have measured the excitation energy and decay mode of two new, T = 1 excited states

in 16F that decay by 2p emission. They have excitation energies of E∗ = 7.67 ± 0.02 MeV

and E∗ = 10.26 ± 0.02 MeV. While we expected the 16FIAS to decay by 2p decay to 14NIAS,

this is highly suppressed by the Coulomb barrier and was not observed. This is likely due to

the reduced phase space and increased Coulomb barrier as compared to the observed cases.

The only possible evidence for 16FIAS is a peak in the γ ray energy spectrum measured in

coincidence with a ET ∼ 0.2 MeV 1p decay. As the main goal of the γ ray array was to

detect the 3.563 MeV γ ray from the decay of 6LiIAS, our electronics were not set up to

95

measure a 10 MeV γ ray. Simulation of the detector response to a 10 MeV γ ray is not

possible without detailed knowledge of these electronics, and therefore no clear conclusion

can be drawn for the γ decay of 16FIAS.

96

Chapter 7

A = 7 IMME

7.1 Background

In the absence of Coulomb forces, if isospin is a good quantum number, the masses of the

members of an isobaric multiplet should be independent of their isospin projections, TZ .

With the addition of the Coulomb force and the proton-neutron mass difference, the masses

can be described by the quadratic Isobaric Multiplet Mass Equation (IMME),

M(T, TZ) = a+ bTZ + cT 2Z . (7.1)

Addition of cubic (dT 3Z) or quartic (eT 4

Z) terms to the IMME measure deviation from the

quadratic form due to isospin-symmetry breaking. However, certain isospin-breaking effects,

such as Thomas-Ehrman shifts and isospin mixing, are entirely accounted for in the b and c

terms [68]. The cubic and quartic terms are generally small and in most cases unnecessary

for quartets and quintets with A ≤ 41 [69]. Among these, only the A = 7, A = 11, and

A = 41 quartets have cubic terms larger than 20 keV [69]. For the A = 7, T = 3/2 quartet

a purely quadratic fit has χ2/n = 4.7, and d = 47 ± 22 keV was required to fit the quartet

[24]. While this value of d is not that large, the deviation from zero is at the 2σ level. The

A=7 quartet is unique amongst multiplets used to test the IMME in that all of it members

97

are resonances. Indeed the narrowest of these resonances is 7Heg.s. with a FWHM of 150

(20) keV [9]. All members are wide enough, and close enough to threshold, that their line

shapes are not adequately described by a Breit-Wigner form. In such cases the resonance

energy is not a well defined quantity, and there are a number of different definitions that can

be employed which can give different values. It is therefore important to use a consistent

definition of the resonance energy for all four members in constructing the IMME.

The A=7 quartet is also important as it is amenable to ab initio calculations. Green’s

Function Monte Carlo (GFMC) calculations suggest that this quartet requires a cubic term

of moderate strength, i.e. d=7.5±2.5 keV [10].

Here we reexamine the A=7 IMME and present new data for the isobaric analog states

(IAS) in both 7Li and 7Be. In addition we refit existing data for 7Bg.s. decay published in

Ref. [24] and subject all three states to a consistent R-matrix analysis [70] of their line

shapes. The line shape of the forth member of the quartet, 7Heg.s., will be taken from

the R-matrix analyses of its n+6He decay channel in Refs. [7, 8]. Note that for all four

members of the quartet, the R-matrix analysis was made with data obtained using the

invariant-mass technique. The fact that all members are resonances also allows us to use the

R-matrix analysis to extract information on the variation of spectroscopic strength across

the multiplet.

Previous measurements of the |TZ | = 1/2 members (7BeIAS and 7LiIAS) of this quartet

were made more than 45 years ago. For 7LiIAS there are two previous measurements, E∗

= 11.19 ± 0.05 MeV from 6Li(n,n′) [71] and E∗ = 11.28 ± 0.04 MeV from 9Be(p,3He)

[72, 73]. For 7BeIAS there exist two measurements, E∗ = 11.00 ± 0.05 MeV from 4He(3He,p)

[74] and E∗ = 11.01 ± 0.04 MeV from 9Be(p,t) [73, 75]. In all these previous studies, the

experimental peaks were fit using Breit-Wigner lines shapes, rather than the R-matrix form,

possibly causing problems for the A=7 IMME studies.

98

7.2 Experimental Method

Data for the decay of 7BeIAS were obtained using the experiment described in Chapter 2.

Data for 7Bg.s. were obtained under almost identical experimental conditions (same beam,

target and detector arrangement) and the invariant mass spectrum was presented in Ref. [24].

7LiIAS states were populated by reactions of a secondary beam of 12Be (1 x 105 pps, 87%

purity) on polyethylene and carbon targets. The decay products were detected in the High-

Resolution Array (HiRA) [19] in 16 ∆E-E [Si-CsI(Tl)] telescopes located 60 cm downstream

of the target. The telescopes were arranged in four towers of four telescopes each, with two

towers on either side of the beam.

7.3 R-matrix formalism

The line shapes of the resonances were fit with the R-matrix formalism [70] which was

originally developed to describe resonance scattering. In this formalism, the scattering wave

for a particular channel c is subdivided into internal and external regions separated by

a surface at relative distance r = ac, called the channel radius. The channel radius is

chosen such that only Coulomb and centrifugal potentials exits in the external region and

therefore the reduced radial wavefunction uc(r) for that channel can be expressed as a linear

combination of regular and irregular Coulomb wave functions. In the isolated resonance

approximation [70], the couplings between the wavefunctions uc(r) and their derivatives

u′c(r) of the different channels in the vicinity of the energy Eλ are given by

uc(ac) =∑c′

γcγc′

Eλ − E[ac′u

′c′(ac′)−Bc′uc′(ac′)] (7.2)

99

where the boundary conditions Bc are the logarithmic derivative of the channel wavefunction

acu′c(ac)

uc(ac)at the energy Eλ. For a resonance that couples to only one channel, this reduces to

acu′c(ac)

uc(ac)= Bc +

Eλ − E

γ2c

, (7.3)

which is just a first-order expansion of the logarithmic derivative in energy about Eλ. From

Eq. (7.2), the S-matrix relating the magnitudes of the incoming and outgoing waves for

the different channels can be derived. For elastic scattering with a channel c, the S-matrix

element can be expressed as

Sc,c = Sresc,c e

(−2iϕc) (7.4)

where the ϕc term is the contribution from hard-core or potential scattering and Sresc,c is the

resonance contribution. Similarly the total phase shift δc, defined by Sc = |Sc,c|e2iδc , has

both a resonant and hard-core contribution, i.e.,

δc = δresc − ϕc. (7.5)

In invariant-mass studies where the resonance is produced by knockout or other inelastic

processes, the potential scattering contribution is irrelevant and so the line shape is assumed

to come just from the resonant contribution, i.e.,

∣∣1− Sresc,c (E)

∣∣2 = Γc(E)

(Eλ +∆tot(E)− E)2 + (Γtot(E)/2)2(7.6)

where the energy-dependent total decay width has contributions from all unbound channels

that couple to the level

Γtot =

(unbound)∑c

Γc(E) (7.7)

100

and the total level shift has contributions from all bound and unbound channels

∆tot =

(unbound)∑c

∆c(E) +

(bound)∑c

∆c(E). (7.8)

The level shifts are ∆c(E) = −γ2c [Sc(E − Ec)−Bc] where Ec is the threshold for that

channel. Given that the isolated level approximation is only valid in the vicinity of the

Eλ, this value should in principle be chosen to be in the center of the resonance region.

However in practice, equivalent line shapes can be obtained with any value of Eλ as long as

the Bc values are adjusted accordingly. However, it is useful to use what has been called

natural boundary conditions defined as Bc = Sc(Eλ − Ec) so that ∆i(Eλ) = 0 and then

Eλ takes the value of the resonance energy (at least in one definition of that quantity [see

Sec. 7.4]). The energy-dependent decay widths are Γc(E) = 2γ2cPc(E−Ec) where Pc is called

the penetrability factor. For unbound states (E > Ec), both P (E − Ec) and S(E − Ec) are

dependent on the regular and irregular Coulomb wavefunction F and G evaluated at the

channel radius. For bound states (E < Ec), P (E − Ec) = 0 and S(E − Ec) is determined

from the Whittaker function at this radius. See Ref [70] for details.

The use of Eq. (7.6) is an assumption which should be justified by reaction theory.

However given the complexity of the reactions in this work, where multiple particles are

removed from the projectiles with possible contribution from different mechanisms, this is not

possible at present. In general one would not expect Eq. (7.6) to be valid at low bombarding

energies where the range of possible excitations is limited by energy conservation. This should

not be a problem in this work. Also in the isolated level approximation, this assumption

(Eq.(7.6)) may not be valid too far from the resonance energy.

The reduced widths can be written as

γ2c = ScC

2c γ

2c (s.p.) (7.9)

where γ2c (s.p.) is the single-particle value of the reduced width, and ScC

2c is the spectroscopic

101

factor for that channel. The quantity Cc is the isospin Clebsch-Gordan coefficient and the

total spectroscopic strength is Sc for isobaric equivalent decays. In the single-particle model,

the overlap function for the channel c is the solution of the Schroedinger equation for the

potential V (r) between the two fragments. If this potential is known and thus acu′c(ac)/uc(ac)

can be calculated, then γ2c (s.p.) can be determined from it’s first order expansion about the

resonance energy [Eq. (7.3)]. Equivalently the single-particle value is also given by [70]

γ2c (s.p.) =

~2

mca2cθ2c , (7.10)

θ2c =ac

2∫ ac0

uc(r)2druc(ac)

2. (7.11)

Many R-matrix analyses ignore the contribution from bound channels to ∆tot. If a bound

channel c is eliminated and we assume the Thomas approximation that ∆c(E) is linear in

energy, then the R−matrix line shape expression Eq. (7.6) still holds but the remaining

partial decay widths for channels c′ must be replaced by effective values

γ2c′(eff) =

γ2c′

1− d∆c/dE. (7.12)

The resonance energy does not change as long as one is using natural boundary conditions.

These γ2c′(eff) values can be used to describe the line shape, however they should not be

used to extract the spectroscopic factor unless corrections for the eliminated channels are

first made. As shown in Ref. [70], for a bound channel:

d∆c

dE= −ScC

2c

∫∞ac

|uc(r)|2dr∫ ac0

|uc(r)|2dr. (7.13)

Therefore the largest correction will be from eliminated channels with large spectroscopic

strength and with energies close to threshold where the wavefunction extends significantly

beyond the channel radius. In this work the most important correction will come from elim-

inating the nucleon+core (Jπ=2+) channels where these conditions hold. However, rather

102

than trying to correct the effective values, we will perform R-matrix fits with these channels

included and thus do not rely on the Thomas approximation.

7.4 Resonance energy definitions

There are many definitions of the resonance energy which give different values for wide

resonances, but coincide in the limit of the resonance width going to zero [76, 77]. For a

single channel, the scattering phase shift undergoes a change of around π radians as the

energy is scanned across the resonance. One could consider the resonance energy as the

value where this change in phase shift is π/2 radians. In the R-matrix formalism where

the total phase shift is conveniently given in terms of the sum of potential scattering and

resonance scattering contributions [Eq. (7.5)], the resonance energy is typically defined as

the energy where the resonant part of the phase shift is π/2 radians, i.e., δres(E(1)r ) = π/2.

This occurs at the energy [70]

E(1)r = Eλ +∆(E(1)

r ) (7.14)

which for natural boundary conditions is just E(1)r = Eλ. Another possible solution is

to choose the energy E(2)r where the total phase shift changes fastest with energy, i.e.,

(d2δ/dE2)E

(2)r

= 0.

For levels which couple to more than one open channel, like the decay of 7BeIAS and

7LiIAS, phase shifts for elastic scattering of the different channels do not always undergo

a change of π radians as one scans the resonance and thus these definitions need to be

generalized. If there are two open channels, then only for the channel with the larger partial

decay width does this phase shift change by π radians. For example, Fig. 7.1 shows the

total neutron and proton phase shifts extracted from our R-matrix analysis of the 7LiIAS

resonance (see later). Here only the dominant proton channel undergoes a phase shift of

close to π radians whereas the neutron channels has little net change in the phase across

the resonance. However, for both channels, Sresc,c is purely real at the energy E

(1)r given by

103

Eq. (7.14) and thus this can be used as a more general definition of this resonance energy.

[MeV]p E-E1 1.5

[r]

δ

1−

0

1

2

3

pδ

nδ

2π

2π

Figure 7.1: (Color Online) Elastic scattering phase shifts for p+6He (δp) and n+6LiIAS (δn)scattering extracted for the 7LiIAS resonance. The energy is relative to the proton thresholdEp.

As for the definition of E(2)r , things are more complex. For two unbound channels,

both channels shown a rapid variation of the total phase shift near the resonance energy

(Fig. 7.1). However, the energy where d2δc/dE2 = 0 can be different for the two channels.

Such a definition is channel specific and therefore not as useful.

Another often-used definition of the resonance energy is the pole of the S-matrix in the

complex energy plane. At this complex energy E(3) = E(3)r − iΓ(3)/2, one has Gamow-state

solutions where the probability decays exponentially in time and there is only an out-going

wave asymptotically. This definition is also valid for multi-channel resonances as all elements

of the S-matrix have poles at the same energy. This pole can be obtained from solving

Eλ +∆tot(E(3))− E(3) − i

Γtot(E(3))

2= 0. (7.15)

104

7.5 Monte Carlo Simulation

In fitting the experimental line shape with the R-matrix parametrization, it is important

to include the experimental resolution. This was determined via Monte Carlo simulations

of the decay in all its aspects and also the detection apparatus. We made two sets of fits.

First we include only the nucleon+core (Jπ = 0+) decay channels and thus we have only one

channel for the 7Bg.s. resonance and two channels each for the isobaric states (see Fig. 7.2).

These fits are comparable to those made for 7He in Refs. [7, 8]. The R-matrix fit parameters

are Eλ and the total spectroscopic strength S0+ . This strength is divided by the isospin

Clebsch-Gordan coefficients into the ratios 2/3 and 1/3 for the proton and neutron decay

channels of 7BeIAS respectively. The ratios are reversed for 7LiIAS decay. The spectroscopic

strengths one obtains in these fits are, in principle, minimum values as they correspond to

effective reduced widths [Eq. (7.12)] due to elimination of bound channels.

To see the effect of including unbound channels, we have also made fits including the

nucleon+core (Jπ=2+) channels which requires the magnitude of S2+ to be specified (see

later). These J=2+ daughter states are wide, especially the 6Be first excited state. We

therefore calculate the energy shift as an integral over the line shapes of the daughters;

∆c(E) = −γ2c

∫w(E0 − Ecen

0 ) [S(E − E0)− S(Eλ − E0)] dE0 (7.16)

where the weight w(E0 − Ecen0 ) is a normalized Breit-Wigner line shape and Ecen

0 is the

centroid of the daughter state. In principle these wide states should also show deviations

from a Breit-Wigner form. We explored the sensitivity to their line shape by using Gaussian

weights with the same FWHM and got equivalent results. For 7Bg.s. and7BeIAS decays,

the line shapes of these wide daughter states extend into the continuum, Fig. 7.2, and so a

similar integral over the line shape will contribute to the decay width Γc(E). It is important

to understand if this influences the predicted line shape significantly and shifts the fitted

resonance energy.

105

For 7B decay, the 6Be daughters subsequently decay to the 2p+α channel. The final 7B

exit channel is thus 3p + α events which can include the Jπ=2+ state of 6Be as well. In

order to determine the detector response we need to simulate the two-proton decay of these

6Be levels. We have extensively studied the decay of both these levels in Ref. [78] where

the correlations between the momentum of all three decay fragments is well described by

Hyperspherical-Harmonic cluster model calculations [78]. We have used Monte-Carlo events

generated by this cluster model for the decay of 6Be fragments in our simulations.

The only significant contributions to the experimental resolution come from the energy-

loss and small-angle scattering of the final fragments in target and the energy resolution of

the CsI(Tl) E detectors. As in our previous work [24], the target effects were treated as

described in Refs. [79, 80]. The CsI(Tl) resolutions, which may be particle dependent, were

determined by fitting a narrow resonance where the exit channels contained the same two

particles as in the resonance of interest. This fitting was made individually for each of the

A=7 resonances using data measured during the same experiment. For the 7LiIAS → p+6He

decay, we used the 6Li(J=3+, E∗=2.186(2) MeV)→ d + α resonance of width Γ=24(2) keV

to fine tune the CsI(Tl) resolution while for the 7BeIAS → p+6Li resonance the 8Be(Jπ=1+,

E∗=17.640(1) MeV, Γ=10.7(5) keV)→ p+7Li was used. Finally for the 7Bg.s. resonance

that decays to the 3p+α channel, we used the 6Beg.s.(Jπ=0+, Γ=92(6) keV)→ 2p+ α decay

for this fine tuning. The extracted CsI(Tl) resolutions were similar in all cases.

7.6 Analysis

7.6.1 General Consideration

Figure 7.2 presents partial energy-level diagrams for the A = 7, T = 3/2 isobar showing only

the levels relevant for this work. In fitting the experimental line shapes we used the channel

radius ac=4 fm for all channels for consistency with the 7He R-matrix analyses in Refs. [7, 8].

For each system, we first fit the invariant-mass spectrum with the nucleon+core (Jπ=0+)

106

Ene

rgy

[MeV

]

4−

2−

0

2

g.s.B7

(T=3/2)-

=3/2πJ g.s.Be6p+

(T=1)

(T=1)+2

α3p+

(a)

Ene

rgy

[MeV

]

0

5

10

15

g.s.Be7

(T=1/2)-

=3/2πJ

(T=3/2) IAS-3/2

g.s.Li6p+(T=0)

+=1πJ

IAS (T=1)+0

(T=1)+2

g.s.Be6n+(T=1)

(T=1)+2(b)

Ene

rgy

[MeV

]

0

5

10

15

g.s.Li7

(T=1/2)-

=3/2πJ

(T=3/2) IAS-3/2

g.s.Li6n+(T=0)

+=1πJ

IAS (T=1)+0

(T=1)+2

g.s.He6p+(T=1)

(T=1)+2

(c)

Ene

rgy

[MeV

]

4−

2−

0

2

g.s.He7

(T=3/2)-=3/2πJ

He+n6

(T=1)

(T=1)+2

(d)

Figure 7.2: (Color Online) Set of levels relevant for the decay of the T = 3/2 members ofthe A = 7 quartet, with (a)-(d) arranged with increasing TZ . The levels are labeled by theirspin-parity (Jπ) and isospin (T) quantum numbers. Observed, isospin-allowed decays areshown with solid red lines, and unobserved, isospin-allowed decays are shown with dashedred lines. Levels to which particle decay is allowed are shown in red. The properties of theA=6 states are listed in Table 7.1

107

Table 7.1: Excitation energy and widths of the A=6 daughter states from Ref. [9].

state 6He 6Li 6BeJ=0+ E∗ [MeV] 0. 3.563 0.

Γ [MeV] 0. 8.2×10−6 0.092(6)J=2+ E∗ [MeV] 1.797(25) 5.366(15) 1.670(15)

Γ [MeV] 0.113(20) 0.541(20) 1.16(6)

channels. The dimensionless single-particle reduced widths θ2s.p. were obtained from the over-

lap functions predicted with the Variational Monte Carlo (VMC) or the Green’s Function

Monte Carlo (GFMC) methods in Ref. [6] using Eq. (7.11). Calculated reduced overlap func-

tions from the two theoretical methods are displayed in Fig. 7.3 for the 7Heg.s. → n+6Heg.s.

decay. The more recent VMC calculations use the Argonne v18 two-nucleon (AV18) [81] and

Urbanna X three-nucleon [82] potentials while the GFMC results used the same two-nucleon

potential but the Illinois-7 three-body interaction (IL7)[83]. Both methods treat this un-

bound channel as being pseudo-bound and thus the asymptotic behavior should be ignored.

The dashed curves are from single-particle calculations with a Wood-Saxon potential V (r)

whose depth was adjusted to give the correct resonance energy, while the radius and diffuse-

ness parameters were adjusted to fit theoretical overlap functions inside the channel radius

(ac=4 fm, vertical dashed line in Fig. 7.3).

These single-particle overlap functions give us an indication of the radial extent to which

the GFMC and VMC overlaps should be trusted. If the channel radius ac were to be shifted

further out from our value of ac=4 fm, then it would be problematic to use the direct

theoretical overlap functions to calculate θ2s.p, especially for the VMC calculation. In such

cases the dashed curves would be a better choice for determining θ2s.p..

The θ2s.p. values for the 7Heg.s. → n+6Heg.s. are 0.584 and 0.700 for the GFMC and

VMC methods respectively. For the VMC method, calculated overlap functions for the

nucleon+core (Jπ=0+) channels of the other isotopes were very similar in shape and sub-

sequently the θ2g.s. values only ranges from 0.700 (n+6Heg.s.) to 0.724 (p+6Beg.s.), a 3%

variation. Such calculations were not available for the GFMC method and so a constant

108

r [fm]0 2 4 6 8 10

]-1

/2u(

r) [f

m

0

0.2

0.4

0.6

0.8

+

VMC, 2

+VMC 0

+GFMC, 0

Figure 7.3: (Color Online) Overlap function predicted by the GFMC and VMC method forthe 7Heg.s. → n+6He channels. For the the VMC methods, results are shown for decaysto the ground (Jπ=0+) and first excited state (Jπ=2+) state of 6He, while only the groundstate is shown for the GFMC method. The Jπ=0+ channel is unbound and the dashedcurves show single-particle overlaps with the correct asymptotic behavior that match withthe theoretical calculations in the interior region (see text for details).

θ2g.s.=0.584 was used for all Jπ=0+ channels.

To gauge the effect of the J=2+ channels we have also performed R-matrix fits with

these channels included. The overlap function for the bound 7Heg.s. → n+6He(2+) channel

predicted with the VMC method is also shown in Fig. 7.3. This overlap is for the p3/2 orbital

only and has a large spectroscopic factor of S2+=2.05. For comparison the shell-model

prediction (code Nushell@MSU[84]) with the Cohen-Kurath effective interaction [85, 86] is

S2+=1.34 (including center-of-mass corrections). There is also a small p1/2 contribution

predicted by the VMC method with strength S = 0.008 that we have ignored. As both

these J=0+ and J=2+ channels are p3/2 in nature, it is not surprising that their overlap

functions have similar shapes in the interior regions. For the n+6He(2+) channel, θ2s.p.=0.685

109

which is very similar to the value for the corresponding J=0+ channel (θ2s.p.=0.700). Results

for corresponding channels in the other isotopes are also similar with a value of θ2g.s.=0.733

obtained for p+6Be(2+). No overlap functions and spectroscopic factors are available for the

GFMC method for the Jπ=2+ channels. For illustrative purposes in this case, R-matrix fits

were performed with a value of θ2s.p.=0.584 (same as for the Jπ=0+ channels) and S2+=2.0.

For each of the three resonances, the fitted energies Eλ = E(1)r are listed in Table 7.2 and

the fitted spectroscopic strengths are listed in Table 7.3. The systematic errors quoted in the

following discussions contain contributions from the uncertainties in the energy and angular

calibrations of the detectors. This was gauged by comparing the invariant mass of known

narrow resonances found in the same data set. See Table 7.4 for a list of these resonances

and their extracted energies.

Table 7.2: Resonance energies and decay widths extracted from the R-matrix analyses ofthe A = 7 cases. The E

(2)r resonance energy is channel dependent and thus values are listed

from both proton (p) and neutron (n) exit channels.

Nuclide resonance energy [MeV] Γ [keV]

E(1)r E

(2)r E

(3)r E

(ENSDF )r R-matrix ENSDF

7He [7] 0.387(2) 0.372(2)(n) 0.372(2) 0.410(8) 132(3) 150 (20)7He [8] 0.430(3) 0.416(3) (n) 0.416(3) 131

7Li 1.288(15) 1.288(15) (p) 1.269(15) 1.266 (30) 191(41) 260(35)1.288(15) (n)

7Be 1.82(2) 1.79(2) (p) 1.78(2) 1.840(30) 376(47) 320(30)1.85(2) (n)

7B 2.037(35) 1.963(38)(p) 1.963(38) 2.013 (25) 599(113) 801(20)

7.6.2 7LiIAS

The isobaric analog state (IAS) in 7Li has two energy- and isospin-allowed decay channels,

Fig. 7.2(c). It can neutron decay to the isobaric analog state in 6Li, which then decays to

the ground-state via a 3.563 MeV γ ray, or it can proton decay to the ground-state of 6He.

Our detectors are not sensitive to neutrons, so only the proton decay channel was observed.

Using the invariant-mass method, the 7Li excitation energy was reconstructed from the

110

Table 7.3: Spectroscopic factors S0+ extracted from the R-matrix fits in this work. Resultsare listed for both the θ2s.p. values obtained from the GFMC and VMC overlap functions andfor S2+=0 and S2+=2.0

Nuclide Sg.s. Ref.GFMC VMC

S2+ = 0 S2+ = 2.0 S2+ = 0 S2+ = 2.07He 0.52(2) 0.99(4) 0.43(1) 0.89(2) [7]7He 0.43(1) 0.82(2) 0.36(1) 0.74(2) [8]7Li 0.39(11) 0.79(20) 0.33(9) 0.79(20) [*]7Be 0.48(9) 0.94(12) 0.40(7) 0.89(12) [*]7B 0.56(14) 0.98(27) 0.45(11) 0.82(20) [*]

detected p+6He events. The excitation-energy spectrum from the polyethylene-target data,

shown in Fig. 7.4, displays a strong peak corresponding to the IAS. The reaction mechanism

responsible for the creation of this state is complex in that it involves the loss of 5 nucleons

from the projectile. There are possible contributions from direct knockout and decays of

other heavier resonances.

In our R-matrix fits we have considered two parametrization for the background under

the peak. The first is just a quadratic, i.e.,

b1(ET ) = a1 + a2ET + a3E2T (7.17)

where the an coefficients were fit simultanously the R-matrix parameters. The second back-

ground form,

b2(ET ) =a1 + a4 (ET − a2)

1 + exp [(ET − a2)/a3), (7.18)

in principle, allows for a more complex variation of the background under the peak. The best

fitted excitation-energy distributions with both background parameterizations and S2+=0 are

shown Fig. 7.4 as the solid curves. However, the two fits are almost identical and cannot

be differentiated in this figure. On the other hand, the fitted backgrounds indicated by the

dotted (b1) and dashed (b2) curve are somewhat different. The choice of background is not

important for extracting the peak energy as the fitted E(1)r values differ by 0.2 keV compared

111

to a statistical error of 3 keV. However, the extracted spectroscopic strength and width of

the peak have a much larger sensitivity to the background form (see later).

From the fitted resonance energy E(1)r , the excitation energy of this level is E∗ = 11.267 ±

0.003 MeV (statistical), with a systematic uncertainty of 0.015 MeV. A fit of the data from

the carbon target agrees within 2 keV. While this is just within the 1σ error bound of the

previous value (E∗ = 11.240 ± 0.03 MeV) [9], the uncertainty is reduced by a factor of two.

A four channel fit with S2+=2 was also made, however, the resulting curve is almost identical

to that obtained with the two-channel fit in Fig. 7.4. The fitted resonance energy E(1)r was

less than 1 keV smaller in value, confirming that the Thomas approximation (Sec. 7.3) is

valid for this resonance. The elimination of bound channels therefore has negligible effect on

the extracted resonance energy. The Thomas approximation that ∆c(E) is linear in energy,

is not expected to hold for energies significantly different from the resonance energy and

indeed small modifications of the intrinsic line shape in the tail region are found, but these

are easily compensated by adjustments to the background contributions in the fit.

From the best fits, the branching ratios for the n+6LiIAS and the p+6Heg.s. decay channels

are 47% and 53% respectively. This result was found for the fits with both S2+=0 and S2+=2.0

and also those with the θ2 parameters obtained from either the VMC and GFMC overlap

functions.

7.6.3 7BeIAS

The level structure for 7Be, Fig. 7.2(b), is very similar to that of 7Li. There are two energy-

and isospin-allowed decays, neutron decay to 6Beg.s. or proton decay to the IAS in 6Li.

However in this case, the proton decay will be followed by γ emission. Figure 7.5 shows the

γ-ray spectrum tagged by the p+6Li channel. The spectrum has been Doppler corrected

event-wise and contains add back of Compton scattered γ rays detected in neighboring

crystals of the CAESAR array. The selection of events which have the 3.563 MeV γ ray

(G1 in Fig. 7.5), identifies T = 3/2 states in 7Be. A spectrum of the excitation energy,

112

E* [MeV]10 11 12

Cou

nts/

140

keV

0

500

1000

He6p+→Li7

Figure 7.4: (Color Online) The excitation energy spectrum for 7Li reconstructed from allp+6He events. The red solid curve is our R-matrix fit with S2+=0 (θ2c taken from GFMC [6]).Two almost identical fits were made with different background contributions. The dottedand dashed curves indicated the fitted background obtained using parametrizations. b1 andb2, respectively.

less the γ-ray energy, from p+6Li events selected by this γ-ray is shown as the solid curve

in Fig. 7.6(a). Previously it was determined in this experiment that 8BIAS fragments,

produced by one-proton knockout, decay by two-proton emission to the IAS in 6Li [14, 87].

Therefore decays of 8BIAS where one proton is not detected will contaminate this spectrum.

By taking the detected p+p+6Li events gated on the 3.563 MeV γ ray and throwing away

one of the protons, a background spectrum can be constructed. The relative magnitude of

this contamination was determined by Monte Carlo simulations of the 8BIAS decay. This

contamination (blue dashed curve) accounts entirely for the peak around 6.5 MeV in Fig.

7.6(a). The majority of the p+6Li yield comes from 2p knockout reactions, with a small

contribution from sequential decay of T = 2 states in 8B. The latter is the second higher-

energy peak in the blue dashed curve. There is also a background under the peak in the

113

[MeV]γE0 2 4 6

Co

un

ts/7

8 ke

V

0

200

400

600

800

1000

B1

G1

Figure 7.5: (Color Online) The Doppler-corrected γ-ray spectrum measured in coincidencewith detected p+6Li events.

G1 gate in Fig. 7.5. The resulting contribution to the excitation-energy spectrum can be

approximated by taking reconstructed p+6Li events in coincidence with the gate B1 in Fig.

7.5. This contribution, scaled by the relative widths of the G1 and B1 gates, is shown as the

red dotted curve in Fig. 7.6(a). The background-subtracted histogram is displayed in Fig.

7.6(b).

A fit of this spectrum with just two unbound channels (S2+=0) and no residual back-

ground is shown as the solid curve in this figure. Fits which included a smooth background

contribution were also made, but gave almost identical R-matrix parameters. The extracted

excitation energy from the E(1)r resonance energy is E∗ = 10.989 ± 0.004 MeV (statistical)

with a systematic uncertainty of 0.015 MeV. This is within the error bar from the previous

value of 11.010±0.030 MeV [9], but reduces the uncertainty by half. Again the four channel

fit (S2+=2.0), produced an almost identical curve to that shown in Fig. 7.6(b) and the same

resonance energy. The fitted branching ratios for the n+6Beg.s and p+6LiIAS decay channels

114

E* [MeV]6 8 10 12 14

Co

un

ts/5

0 ke

V

0

100

200γLi+p+6

backgroundγ backgroundIASB8

(a)

E* [MeV]6 7 8 9

Co

un

ts/5

0 ke

V

0

50

100

150

200(b)

Figure 7.6: (Color Online) (a) Reconstructed minimum excitation-energy spectrum (missingthe γ energy) for the p+6Li events in coincidence with a 3.563 MeV γ ray (G1 in Fig. 7.5).The blue dashed histogram indicates a contamination from 8B+p+p+γ events where oneproton misses the array. The red dot-dashed histogram is a background from the γ-ray gate(B1 in Fig. 7.5). (b) The points are the background-subtracted excitation-energy spectrum.The red solid curve is a simulation of the decay from 7BeIAS using a R-matrix line shape (θ2ctaken from GFMC [6]) with the experimental resolution included.

115

are 9% and 91%, respectively.

7.6.4 7Bg.s.

The 3p+α invariant-mass spectrum in Fig. 7.7 was published previously by us in Ref. [24].

The particles were produced following interactions of an E/A=70-MeV 9C beam with a 9Be

target and the final decay fragments were detected in the HiRA array configurated in almost

the same arrangement as for the 7BeIAS experiment.

[MeV]TE0 2 4 6 8 10

Cou

nts/

92 k

eV

0

500

1000

1500

α3p+→B7

Figure 7.7: (Color Online) Invariant mass of 7B from all detected 3p+α events. TwoR-matrixfits with S2+=0 are shown as the red solid curve using both background parameterizations.These two fits are almost identical and cannot be differentiated, but their fitted backgroundsare somewhat different. The blue dotted and blue dashed curves were obtained with the b3and b4 parameterizations, respectively.

This spectrum has been corrected for the contribution from 8Cg.s. →4p+α events where

only three of the protons are detected. This correction involving taking detected 4p+α events

and throwing one of the protons away. The resulting 3p+α invariant-mass spectrum was

scaled by the number of such events predicted in our Monte Carlo simulation of 8Cg.s. decay

116

and subtracted from the raw data [24]. This correction did not change the peak location

significantly which largely determines the resonance energy. In extracting the uncertainties

for R-matrix parameters we have considered a ±10% uncertainty in the normalization of

these pseudo 3p+4α events. In addition to removing the contribution from 8Cg.s. peak, we

have also considered the effect of removing the contribution from higher excitation 4p+α

events that form a tail on the ground state. This was not done in Ref. [24]. This tail is

rather flat and this subtraction does not make a significant change in the fitted resonance

energy.

Even after subtracting these background components, the resulting spectrum still shows

a considerable residual background under the 7B resonance. Indeed this residual background

is not smooth as the total counts drops sharply for the decay energies ET below 2 MeV. The

nature of this background with its threshold-like feature is unclear. As it extends below the

7B ground-state peak and then drops sharply, it cannot be an excited 7B resonance. To

explore the sensitively of extracting the R-matrix parameters, we have used two different

parameterizations for the residual background which can produced quite different shapes.

The first parametrization,

b3(ET ) =a1 + a2ET + a3E

2T

1 + exp [−(ET − a4)/a5](7.19)

is a quadratic polynomial with an inverse Fermi function to produce the threshold-like fea-

ture. The second parametrization is

b4(ET ) = a11

1 + exp [−(ET − a2)/a3]

1

1 + exp [(ET − a4)/a5]

+ a61− a7 ∗ (ET − a4)

1 + exp [−(ET − a4)/a5](7.20)

The best-fit curves with these two background parameterizations and a single channel in

the R-matrix prescription (S2+=0) are almost identical and are shown by the solid curve in

117

Fig. 7.7. While these two curves are the same, their decomposition into peak and residual

background components are somewhat different. The fitted background contributions are

indicated by the dotted [b3(E)] and dashed [b4(E)] curves in this figure. The fitted resonance

energy is E(1)r =2.036±0.009 MeV (statistical) with a systematic error of 26 keV which the

includes the uncertainties from the various background contributions.

For this resonance, the p+6Be(2+) channel is partially unbound [Fig. 7.2(a)] and would

also contribute to the detected 3p+α events. Therefore it is important to understand how

it affects the shape of the 7B invariant-mass peak. We therefore refit the experimental

data with a two-channel R-matrix expression with S2+=2.0 and the resulting fitted curve

and resonance energy are almost identical to that obtained with just one channel (S2+=0).

Fig. 7.8 compares the fitted intrinsic lines shapes attained from the one and two-channel fits

with the b4(ET ) background parameterization. The two line shapes are essentially identical

except for the high-energy tail region with the same FWHM. Our fits are insensitive to the

magnitude in the tail region as any change in the tail is easily compensated by changes in

the background contribution. Thus the main effect of including the decay to the 2+ state,

similar to that obtained for the other A=7 resonances, is to modify the spectroscopic factor

extracted for the main decay channel. We would like to point out that the extracted width

is smaller, but with a larger uncertainty, than the ENSDF value (see Table 7.2). The latter

was in fact from our previous analysis [24] that we now know had a fitting error.

7.7 Isobaric Multiplet Mass Equation

From the R-matrix fits in Sec. 7.6 we have learned that the extracted resonance energies are

independent on whether the nucleon+core (Jπ=2+) channels are included or not. Up until

this point, we have concentrated on the E(1)r definition of the resonance energy. Table 7.2

compares these values to those obtained from from the poles of the S matrix (E(3)r ) via

Eq. (7.15). Not unexpectedly, the largest difference between these two resonance energies of

118

[MeV]g.s.Be6p-E

0 2 4

rela

tive

stre

ngth

0

2

4

6

=0+2S

=2.0+2S

B7

(1)

rE

(3)

rE

Figure 7.8: (Color Online) R-matrix line shape for 7B with (without) the contribution ofthe Jπ = 2+ state included shown as the red dotted (blue solid) curve.The location of the

resonance energy input to the R-matrix (E(1)r ) is shown in green, and the resonance energy

from the pole of the S-matrix (E(3)r ) is shown in magenta.

74 keV is for the widest resonance 7Bg.s.. The location of these two resonance energies are

indicated on the Fig. 7.8 which displays the fitted intrinsic line shape. The value obtained

from the pole of the S matrix (E(3)r ) is located just below the maximum of the line shape

while the other value E(1)r sits just above. This difference varies quite smoothly across the

quartet and therefore it can easily be accounted for in the quadratic IMME.

The experimental masses of the T = 3/2 quartet obtained from the E(1)r definition are

listed in Table 7.5. In this case the mass of 7He is taken from Ref. [8]. The most recent

mass evaluation for this nucleus is based on the E(1)r resonance energy [67]. Figure 7.9

shows a comparison between the residuals of a quadratic fit with the previous masses (closed

circles) [88] and a quadratic fit with the new masses determined ether with the E(1)r (closed

triangles) or the E(3)R (open triangles) definitions. The error bars used in the fit are estimates

119

Table 7.4: Comparison between the measured excitation energies (E∗exp) and the literature

values (E∗ENSDF ) [9] for narrow resonances used to estimate the systematic uncertainties in

the two experiments. The deviation is given as ∆E∗ =∣∣E∗

exp − E∗ENSDF

∣∣. For the 7Bg.s.

experiment, we list equivalent results for the total decay energy ET of 6Be.

Channel Jπ E∗exp E∗

ENSDF ∆E∗

(MeV) (MeV) (keV)7LiIAS

6Li→ d+α 3+ 2.1930 (3) 2.186 (2) 7 ± 27Li→ t+α 7/2− 4.6315 (7) 4.630 (9) 1 ± 9

8Be→ p+7Li 1+ 17.643 (1) 17.6400 (10) 3.0 ± 1.410B→ α+6Li 3+ 4.7827 (13) 4.7740 (5) 8.7 ± 1.49B→ p+α+α 5/2− 2.3610 (11) 2.345 (11) 16 ± 1110B→ p+9Be 2+ 7.4829 (50) 1 7.470 (4) 13 ± 6

2− 7.479 (2) 3 ± 57BeIAS

6Li→ d+α 3+ 2.1850 (2) 2.186 (2) 1 ± 28Be→ α+α 0+ 0.0014 (3) 0.00 (7) 1.4 ± 0.38B→ p+7Be 1+ 0.7692 (3) 0.7695 (25) 0.3 ± 2.5

3+ 2.3207 (7) 2.320 (20) 1 ± 208B→ p+p+6Li 0+ 10.6149 (2) 10.619 (9) 4 ± 99B→ p+α+α 3/2− 0.007 (1) 0.0000(9) 7 ± 1

5/2− 2.358 (2) 2.345 (11) 13 ± 118Be→ p+7Li 1+ 17.6470 (8) 17.6400 (10) 7.0 ± 1.3

EexpT EENSDF

T ∆ET

(MeV) (MeV) (keV)7Bg.s.

6Be→ p+p+α 0+ 1.364(6) 1.371(5) 7±11

of the systematic uncertainty. The statistical errors are about a factor of three smaller. As

expected the residuals obtained using the two definitions of the resonance energy are very

similar. The IMME coefficients from the two fits are listed in Table. 7.6. It is clear that the

quadratic fit reproduced the masses very well; with χ2/n=0.66 (E(1)r ) and χ2/n=0.38 (E

(3)r ).

A fit with a cubic dependence yields coefficients for the T 3Z term of d=19.5±13.8 keV for

both definitions, consistent with zero within the 2σ limit. As mentioned previously, GFMC

(AV18+IL7) calculations, where charge independence breaking interactions are included,

predict a value of d= 7.5±2.5 keV [10]. The other coefficients from this calculations are also

listed in Table 7.6 and compared to the cubic fit coefficients from this work. They predict

120

a slightly larger TZ dependence than our two fits. However the predicted cubic term is still

consistent with our experimental result. The largest contribution to d in these calculations

is from the expansion of the nucleus with decreasing TZ . More accurate determination of the

experimental masses are needed to confirm if a non-zero d of this magnitude is required. A

quadratic fit of the experimental masses using instead the mass of 7He from Ref. [7] provides

a very similar fit with χ2/n=1.23 (E(1)r ) and χ2/n=0.84 (E

(3)r ).

Table 7.5: Mass excesses for the A = 7, isospin T = 3/2 quartet using the E(1)r resonances

energies and the coefficients from the quadratic fit.

Nucl. TZ Mass excess a,b,c(keV) (keV)

7B -3/2 27700 (35) a = 26406 (14)7Be -1/2 26758 (20) b = -557 (11)7Li 1/2 26169 (15) c = 213 (10)7He 3/2 26050 (8) χ2/n = 0.66

Table 7.6: Comparison of the coefficients obtained from the cubic fits to the quartet massesfor the two definitions of resonance energy (E

(1)r , and E

(3)r ) and from the GFMC calculations

of Ref. [10].

Coef. E(1)r E

(3)r GFMC

b -594(28) -574(28) -591(8)c 206(11) 198(11) 237(8)d 19(14) 19(14) 7.5(2.5)

We have also considered the sensitivity of results to the choice of the channel radius. In

addition to the fits with ac=4 fm, we repeated the fitting with ac=5.5 fm. The θ2s.p. values

were determined from the dashed-curve in Fig. 7.3 with the correct asymptotic behavior.

The fitted resonance energies shifted by less than 2 keV, much less that our experimental

uncertainties and therefore this can be ignored. In addition the fitted resonance energies

were the same for the VMC and GFMC values of θ2s.p..

We have also looked at the result with the second definition of resonance energy E(2)r

based on the derivative of the phase shift which is problematic in that it may have a channel

dependence for the isobaric analog states (Sec. 7.4). For the 7LiIAS where the decay branches

121

for the neutron and proton are similar (Sec. 7.6.2), the deduced values of E(2)r are identical,

see Table 7.2. However for the 7BeIAS, the two values differ by 54 keV. If we take the value

for the dominant p+6LiIAS channel and construct the IMME with this and the other E(2)r

values we also get small residuals with a χ2/n of 0.98 which is only slightly larger than those

for the other definitions of the resonance energy.

ZT1− 0 1

) [k

eV]

Z2+

cTZ

M-(

a+bT

∆

100−

50−

0

50

100 B7 Be7 Li7 He7

Literature Values

r(1)

E

r(3)

E

Figure 7.9: (Color Online) Deviation from the fitted quadratic form of the IMME for the

lowest T = 3/2 states in the A = 7 isobar. The data points for the E(1)r and E

(3)r definitions

are shifted by 0.1 and 0.2, respectively, along the abscissa for clarity.

7.8 Variation of Spectroscopic Strength

To the extent that isospin is a good quantum number, the spectroscopic strength S0+ should

be constant across the quartet. Figure 7.10 shows the variation of the fitted S0+ values as a

function of the isospin projection TZ . Results are shown for S2+=0 (open-squares) and for

122

ZT2− 1− 0 1 2

+ 0S

0

0.5

1

VMC

(b)

+ 0S

0

0.5

1

GFMC

g.s.B7IASBe7

IASLi7g.s.He7

(a)

Figure 7.10: (Color Online) Ground-state spectroscopic factor as a function of isospin pro-jection (TZ) using θ2c from (a) Green’s Function Monte Carlo and (b) VMC. The open bluedata points have S2+ = 0 and the solid magenta data points have S2+ = 0.75. The top- andbottom-pointing triangles are from [7] and [8], respectively. The horizontal lines show thepredicted spectroscopic factors for 7Heg.s. from the respective models.

123

S2+=2.0 (filled circles) and for the θ2s.p. determined from the GFMC [fig. 7.10(b)] and VMC

[Fig. 7.10(b)] overlap functions. The effect of including the decays to the Jπ=2+ daughters

is very important. The addition of the bound channels increases the fitted S0+ spectroscopic

strength by roughly a factor of two for all resonances. This is of course dependent on our

assumed values of S2+ . However, the addition of bound channels will always increase the

fitted spectroscopic strength of the unbound channel. At the channel radius the bound

channel mixes with the unbound channels, pulling decay strength away from the unbound

channel, therefore the spectroscopic strength of the unbound channel must increase in order

to reproduce the observed decay width.

For 7Heg.s., we include spectroscopic strengths extracted from the fitted reduced widths

γ2c listed in Ref. [7] (up-facing triangles) and Ref. [8] (down-facing triangles) from single-

channel R-matrix fits. The results for S2+=2 in Fig. 7.10 and Table 7.3 were obtained from

fitting the intrinsic lines shapes obtained with the single-channel fits. The authors of these

two studies use a different method to extract the spectroscopic factor from the reduced width

involving a comparison of the fitted reduced width to that obtained for the 5He resonance

which is assumed to be a single-particle resonance. This procedure will not account for the

difference in eliminated channels for the two resonances. However their spectroscopic factors

are not too different from our S2+=0.0 value with S0+=0.619(22) [7] and 0.512(18) [8].

Within the experimental uncertainty, the S0+ strength is roughly consistent with a con-

stant value if one compares the same S2+ values and same overlap functions. However, the

larger errors bars makes more detailed conclusions impossible.

The present fits show that we cannot extract the S0+ spectroscopic strength without

knowledge of the value for the Jπ=2+ channels. However we can look for consistency with

the VMC method as both S0+ and S2+ are predicted. The connected small data points in

Fig. 7.10(b) show the predicted spectroscopic strength S0+ obtained from the VMC model.

All values are very similar with magnitudes of ∼0.526. For the GFMC method only the

7Heg.s. result of S0+=0.532 has been calculated and so this value is plotted as the horizontal

124

line in Fig. 7.10(a). For comparison the shell-model value of S0+ is 0.696. In Fig. 7.10(b),

the solid data points obtained with the VMC value of S2+=2 are 30% to 80% larger than

VMC predictions. For the GFMC method no value of S2+ has be predicted yet. To show

consistency, the value of S2+ would have to be small as already the results with S2+=0 are

close to the open data points in Fig. 7.10(a).

ZT2− 1− 0 1 2

[keV

]Γ

0

200

400

600

800

only+, 0(3)Γ

only+FWHM, 0+,2+, 0(3)Γ

+,2+FWHM, 0

g.s.B7IASBe7

IASLi7g.s.He7

Figure 7.11: (Color Online) Comparison of the width of the T = 3/2 resonance for A = 7as a function of TZ from the FWHM of the R-matrix line shape and extracted from theS-matrix. The contribution of including the Jπ = 2+ is also shown. The dotted line is aquadratic fit.

The spectroscopic factors constrain the total width of these resonance. Figure 7.11 com-

pares the widths obtained four ways. The decay width can be determined from the FWHM

of the fitted intrinsic line shape or from Γ(3), the imaginary part of the pole of the S-matrix

(see Sec. 7.4). In addition for each of these, we consider the results obtained for the fits with

S2+=0 and 2.0. However, as can be seen from this figure, all ways of determining the width

of the resonance yield consistent results. The widths are well fit with a quadratic dependence

on TZ shown as the dotted curve. If a higher-order polynomial was needed in fitting these

widths one expects that this would lead to disagreement with the quadratic IMME using the

125

E(3)R definition.

7.9 Summary

In this work we have reexamined the A = 7, T = 3/2 quartet. New measurements of isobaric

analog states of 7Be and 7Li, with better statistics and better resolution than previous work,

have reduced the uncertainty on their resonance energy by at least a factor of two in both

cases. As the intrinsic widths of all four members of this quartet are large, the resonance

energy is not a well defined quantity. In order to make a fair comparison we have analyzed all

four members with a consistent R-matrix analysis. Regardless of the chosen definition of the

resonance energy, with the new measurements for the |TZ | = 1/2 members, the masses are

well reproduced by the quadratic form of the isobaric analog mass equation. Furthermore we

have extracted the spectroscopic strengths across the multiplet. We find that the extracted

spectroscopic factors for decay to the Jπ = 0+ daughter states are highly dependent upon

the assumed strength for configurations associated with the first-excited Jπ=2+ states of

these daughters. The latter configurations are bound except for the proton-rich members

where they are partially unbound. The strength of both these configurations obtained from

the Variational Monte Carlo method were not consistent with our R-matrix analysis of

experimental lines shape of the isobaric quartet.

126

Chapter 8

Assorted States

8.1 9C

Ene

rgy

[MeV

]

0

2

4

6

g.s.C9

)-(T=3/2,3/2

-1/2

-5/2

g.s.B8p+)+(T=1,2

+1

+3

g.s.Be7p+p+)-(T=1/2,3/2

-1/2

-7/2

Figure 8.1: Partial level scheme for the proton decay of 9C and 8B. Levels are labeled bytheir spin and parity (Jπ) when known. New levels, widths, or decay modes are plotted inmagenta.

Excited states of 9C were populated through inelastic scattering of a 9C beam with a 9Be

target. A partial level scheme for the one- and two-proton decay of 9C is shown in Fig. 8.1.

127

Previously known information (energy, width and decay modes) is plotted in black for 9C

and the low-lying states in the daughter nuclei. The ground state is the only particle-bound

state in 9C, with the first-excited state more than 1 MeV above the proton-decay threshold.

The invariant-mass spectrum of 9C from all detected p+8B events is displayed in Fig. 8.2.

The two prominent peaks correspond to the known first- and second-excited states with Jπ

= 1/2− and 5/2− respectively. Both states decay by one-proton emission to the Jπ = 2+

ground state of 8B.

C) [MeV]9 (TE0 2 4 6

Cou

nts

0200400600800

10001200140016001800 B8p+

Sum-1/2-5/2

Background

Figure 8.2: Invariant-mass spectrum of 9C from all detected p+8B events.The blue dashedand green dotted lines are R-matrix simulations for the 1/2− and 5/2− states, each curvehas been scaled down by a factor of two for clarity. The orange dot-dashed line is oneparameteriza0.0tion of the background. The sum of the background and two simulations isplotted as the red solid line.

0. +

In order to extract the level energy and width for each state, the invariant-mass spectrum

was fit using the R-matrix, described in Chapter 7. The red solid curve in Fig. 8.2 is the

best fit with the excitation energy for the first-excited state fixed at the known value (2.218

128

B) [MeV]8E* (0 2 4 6 8

C)

[MeV

]9

E*

(

0123456789

10

B)8(π= JG1

+1G2

+3

(a)

C) [MeV]9E* (0 5 10 15

Co

un

ts

0

100

200

300

400

500

600

700

800(b)

Figure 8.3: (a) Excitation energy spectrum for 2p+7Be events plotted as a function of theexcitation energy for the intermediate (8B) for each p+7Be combination. (b) Excitationenergy spectrum for 9C from all detected 2p+7Be events (black). The red and blue spectraare the excitation energy spectrum for 9C gated on gates G1 and G2, respectively.

129

(11) MeV [9]) and the energy for the second-excited state extracted from the fit. The fit

includes the individual R-matrix simulations of each peak (blue dashed and green dotted

lines respectively) as well as a smooth background (yellow dot-dashed line). The background

is an inverse Fermi function multiplied with a quadratic to best describe a background with

a sharp cutoff, similar to the background chosen for 7B in Chapter 7. The line shapes

of the individual peaks are taken from R-matrix line shapes of a given decay energy and

width that were input into a Monte Carlo simulation to account for detector resolution and

bias. The amplitudes of the line shapes and the parameters for the background were then

simultaneously fit for each decay energy and width pair to extract the over-all best fit to

the data. The width extracted for the first-excited state is Γ = 52 ± 11 keV. The excitation

energy of the second-excited state was found to be E∗ = 3.549 ± 0.020, with an intrinsic

width of Γ = 673± 50 keV. The extracted width of the first-excited state is much lower than

the value of Γ = 100 ± 20 keV obtained from (3He,6He) transfer in Ref. [89]. Recent ab

initio calculations with the Variational Monte Carlo method predict the width to be Γ =

102 ± 5 keV [90], which is consistent with the transfer work, but totally inconsistent with

our measurement. However more recent results from the Shell Model with the source-term

approach (STA) predict a width of ΓSTA = 53 keV, which reproduces our result [91]. The

standard approach to the Shell Model also agrees with our result, ΓSM = 58.7 keV [91].

At E∗ = 1.436 (18) MeV, 9C becomes unbound with respect to two-proton emission. The

black histogram in Fig. 8.3 (b) shows the invariant-mass spectrum from all detected 2p+7Be

events. A wide, asymmetric peak at around 5.5 MeV can be seen. Any excited states of

9C that 2p decay will decay sequentially through the excited states of 8B. Therefore we

can reconstruct the excitation energy of the 8B intermediate, and determine the decay path

through which each state in 9C de-excites, Fig 8.3 (a). Two peaks can be seen, corresponding

to the 1+ and 3+ excited states in 8B. Gating on these two states, gates G1 and G2, pull

out two different peaks from the 9C invariant-mass spectrum. The G1 gate (red histogram)

is associated with a state in 9C at 4.40 (2) MeV, decaying through the 1+ state in 8B, and

130

the G2 gate (blue histogram) is associated with a state at 5.69 (2) MeV, decaying through

the 3+ state. The 4.40 MeV state was seen previously from (3He,6He) transfer reactions,

where they reported only the excitation energy (4.3 MeV) [92]. While no Jπ for either state

is known, nor are the analogs known in the mirror nucleus 9Li, we can infer that the lower

energy state is low spin, and likely J = 1/2+ because it decays predominantly through the

1+ state and not the 2+ ground-state in 8B. Similarly the higher energy state is likely high

spin, J ≥ 7/2, because it decays mostly through the 3+ state.

B) [MeV]8E* (0 0.5 1 1.5 2 2.5 3 3.5 4

Co

un

ts

0

2000

4000

6000

8000

10000

Figure 8.4: Excitation energy spectrum of 8B from all p+7Be events (black solid line). Thepeaks corresponding to the decay of the 1+ state are fit with the sum of two Gaussian fitsand a linear background (dashed line).

8.2 8B

The two-proton decay of 8B was discussed previously in Chapter 3, and the one-proton

decay to high-lying excited states of 7Be that decay to 3He+α was published previously by

131

our group [24]. In this section I will discuss the one-proton decay to the particle-bound

states of 7Be. A partial level scheme for 8B can be seen in Fig. 8.1. The energy and width

of the first two excited states in 8B have been known for a long time [9], however their

decay modes were not previously reported. The 1+ and 3+ levels are both unbound with

respect to proton decay to the ground state (3/2−) and first-excited state (1/2−) of 7Be.

The first-excited state of 7Be decays via emission of a 429 keV γ ray to the ground-state.

The invariant-mass spectrum for 8B from all p+7Be events can be seen in Fig. 8.4. The

two prominent peaks at E∗ = 0.769 (25) and 2.320 (20) MeV correspond to the first two

excited states of 8B. A smaller peak at 0.340 MeV arises from events where we populate

the first-excited state in 8B and it decays by one-proton emission to the 429.08 (10) keV

first-excited state in 7Be which γ decays. A γ-ray spectrum, gated on this small peak, can

be seen in Fig. 8.5. The blue dashed line corresponds to one estimation of the background.

The background was generated by gating on events in the 9B → p+2α channel, which has

no bound, excited states that produce γ rays. We can extract a branching ratio for the

decay of the 1+ state to either the 3/2− ground state or 1/2− first-excited state in 7Be. By

fitting both peaks and extracting the areas, and correcting for the efficiencies of detection,

the branching ratio was found to be 1.1 % to the 1/2+ excited state and 98.9 % to the 3/2+

ground state of 7Be.

8.3 17Na

States in 17Na can be populated with a 17Ne beam by charge exchange reactions (exchanging

a neutron for a proton) with the target. Prior to this work it was known that 17Na was

unbound but its continuum structure had been unexplored. The decay-energy spectrum for

17Na → 3p+14O is shown in Fig. 8.6. There is a peak in the spectrum located at ET = 4.85

(6) MeV that sits on top of a large background. This background is largely due to 15O nuclei

which are misidentified as 14O. However no strong resonances were detected in the 3p+15O

132

[MeV]γE0 0.2 0.4 0.6 0.8 1 1.2

Co

un

ts

0

20

40

60

80

100

120

140

160

Figure 8.5: The black solid histogram is the γ-ray energy spectrum gated on the smallpeak in the 8B invariant-mass spectrum. The blue dashed histogram is a CAESAR γ-raybackground.

decay channel, so the background in this spectrum is featureless. The mirror nucleus, 17C,

has a 3/2+ ground state, and 1/2+ and 5/2+ excited states at 210 and 331 keV respectively

[93]. For a 3p decay at 4.85 MeV, Monte Carlo simulations of the device resolution predict

the full width at half maximum (FWHM) of the reconstructed energy to be 540 keV. The

FWHM of the detected peak is 1150 keV. Therefore this peak is either a wide state, or some

mixture of all three states (including the ground state), with the average < ET > = 4.85 (6)

MeV.

From the average decay energy, an upper limit for the mass excess of 17Na is ∆17Na ≤

34.72 (6) MeV (blue arrow in Fig. 8.6). The mass excess predicted from systematics is

∆17Na = 35.346 (23) MeV (red arrow in Fig. 8.6) [94]. The measured upper limit on the

mass excess is lower than the systematics by ∼ 600 keV, however this may be due to a

Thomas-Ehrman shift of the 1/2+ excited state, which would bring its energy down relative

to its position in 17C. Shell model calculations suggest that the 1/2+ state has two of the

133

[MeV]TE0 1 2 3 4 5 6 7 8 9 10

Co

un

ts

0

5

10

15

20

25

30

35

40

45

Figure 8.6: Decay energy spectrum of 17Na from all detected 3p+14O events. The bluearrow marks the average energy of the peak, and the red arrow marks the ground-stateenergy predicted from mass systematics.

valence d5/2 nucleons coupled to 0+ and the third valence nucleon in the second s1/2 orbit

[95]. If that third nucleon is a proton, the radial wavefunction will be expanded relative

to the wavefunction for a neutron in that orbit, and the energy of the state will be lower.

It is possible that the s1/2 orbit has moved far enough down in energy that this peak is

completely the 1/2+ state, and therefore the ground-state is 1/2+ instead of being 3/2+ as

it is in the mirror nucleus. To provide further details, a higher statistics experiment with

better resolution is needed.

134

8.4 17Ne

With a 9C beam we were able to populate 1- and 2-proton decaying states in 9C through

inelastic excitation, Sec. 8.1. With a 17Ne beam we are also able to populate 2-proton

decaying excited states in 17Ne through inelastic excitation. The continuum states of 17Ne

up to E∗ = 6.5 MeV were studied almost 20 years ago through the 20Ne(3He,6He)17Ne

reaction [96]. They were able to extract energies, spins, and parities for a large number of

excited state, but no information about how they decay. In this section we will re-examine

the Jπ = 5/2− second-excited state in 17Ne.

MeV

0

1

2

3

4

g.s.Ne17

-0.0 1/2

-1.29 3/2

-1.76 5/2

+2.65 5/2

-3.55 9/2

F+p16

-1.47 0

-1.66 1

O+p+p15

-0.93 1/2

Figure 8.7: Partial level scheme for the 2p decay of 17Ne. States are labeled by their spin,parity, and energy relative to the ground state of 17Ne.

Figure 8.7 shows a partial level scheme for 17Ne. While the 3/2− first-excited state is

above the 2p threshold, it has been shown to decay exclusively by γ-ray emission to the

ground-state through the non-observation of the 2p decay [97]. They measured the 2p decay

branch of the 5/2− state, and concluded that based on the lifetimes from a simple barrier

135

penetration calculation all of the strength had to go through the 0− (τ = 1.4 fs) intermediate

and not the 1− (τ = 300 ps). This is reasonable given the decay energy to the 1− is only

100 keV and the angular momentum in both possible decays is ℓ = 2. From the momentum

correlations in the Jacobi coordinate system, see Chapter 1, we can distinguish between these

two possible decay paths directly.

E* [MeV]2 3 4

Co

un

ts

0200400600800

10001200140016001800

Figure 8.8: Excitation energy spectrum of 17Ne from all detected 2p+15O events. The solidred curve is an R-matrix fit to the 1.7 MeV (Jπ = 5/2−) second-excited state of 17Ne witha quadratic background (red dashed line). Peaks corresponding to the Jπ = 5/2+ and 9/2−

states are seen at higher energy.

The invariant-mass spectrum for 17Ne, Fig. 8.8, shows three peaks corresponding to E∗ =

1.764 (12), 2.651 (12) and 3.548 (20) MeV states with Jπ = 5/2−, 5/2+ and 9/2− respectively,

all seen in [96]. No evidence for the 1.908 (15) MeV 1/2+ state is seen. An R-matrix fit

of the 5/2−, filtered by detector acceptance and resolution via Monte Carlo simulation, was

performed with a simple quadratic form assumed for the background contribution. The

extracted excitation energy is E∗ = 1.77 (2) MeV which is consistent with the result from

transfer [96]. The lifetime of this state predicted from [97] is τ = 1.4 fs, which corresponds to

an intrinsic width of 0.23 eV, far smaller than our detector resolution. Indeed the R-matrix

136

fit of this state with no intrinsic width is consistent with zero width and thus the lifetime

quoted above.

In the Jacobi Y energy variable (Ecore−p/ET ), the expected signature of a sequential decay

is a double peaked spectrum, with one peak corresponding to the decay energy in each step.

Figure 8.9 (a) shows the Jacobi Y energy distribution for the 2p decay of this state. It has two

peaks, one at ∼ 0.35 and one at ∼ 0.65, which is roughly consistent with the decay through

the 0− intermediate. The energy-angular correlations (Fig. 8.9 (b)) in the Jacobi Y system

for this decay exhibit the double-ridge feature seen for sequential decay. Two R-matrix

simulations were performed for the decay either through the 0− or the 1− intermediates,

with the energy-angular correlation spectra in Fig. 8.9 (c) and (d) respectively. Their 1-D

energy projections are plotted as the red solid and blue dashed curves in Fig. 8.9 (a). The

Ecore−p/ET distribution is consistent with the simulation through the 0−, which confirms the

inference of [97].

The most surprising result is the outcome of the simulation of the decay through the 1−.

In that case the decay energy of the two steps are 100 keV and 730 keV, with Γ < 40 keV

for the 1−. The naive expectation would be for there to be two peaks, one at ∼ 0.12 and

one at ∼ 0.88, however only a single peak at 0.5 is seen. To understand why this is the

case, one needs to look at the R-matrix line shape input into the Monte Carlo simulation.

Figure 8.10 shows the line shapes as a function of the energy of the first proton for the decay

through the (a) 0− and (b) 1−. The black curves are the line shapes of the second proton

(p2) as a function of the energy of the first proton (p1), which are peaked at the energy of

the p+16F resonance. The red curves are the barrier penetration factors (Pℓ(p1)) for the

first protons, which fall off sharply to zero with decreasing energy. The product of these two

functions is shown in blue. For the 0− intermediate the peak of the resonance falls in an

energy region where the barrier penetration factor isn’t changing that rapidly, and therefore

the line shape in the resonance region is not altered significant. However for the 1− the

resonance is in a region where the barrier penetration factor is changing quite rapidly. The

137

result is the peak of the resonance is highly suppressed relative to the tail, and the decay

largely happens through the long tail of the resonance line shape. This is a common feature

for near-thresholds decays, often called a “ghost peak”, and has been discussed in more detail

for the α decay of 8Be [98]. The exact shape depends on the assumed width of the 1− state,

in this case the value of Γ = 40 keV was taken. If this was significantly more narrow, then

the Ecore−p distribution would again have two peaks.

138

T/Ecore-pE0 0.2 0.4 0.6 0.8 1

Cou

nts

0200400600800

10001200140016001800200022002400(a)

T/Ecore-pE0 0.2 0.4 0.6 0.8 1

) kθC

os(

1−0.8−0.6−0.4−0.2−

00.20.40.60.8

1(b)

T/Ecore-pE0 0.2 0.4 0.6 0.8 1

) kθC

os(

1−0.8−0.6−0.4−0.2−

00.20.40.60.8

1(c)

T/Ecore-pE0 0.2 0.4 0.6 0.8 1

) kθC

os(

1−0.8−0.6−0.4−0.2−

00.20.40.60.8

1(d)

Figure 8.9: (a) Jacobi Y energy variable for the 2p decay of 17Ne second-excited state. Redsolid (blue dashed) curves are Monte Carlo simulations for the decay through the groundstate (first-excited state) of the 16F intermediate. (b) Jacobi Y energy-angular correlationsfor 17Ne second-excited state. (c) and (d) are Monte Carlo predictions for the decay throughthe ground and first-excited state of 16F respectively.

139

E1[MeV]0 0.2 0.4 0.6 0.8

Yield[arb]

12−10

10−10

8−10

6−10

4−10

2−10

1

210

- 1/2→

- 0→

-(a) 5/2

E1[MeV]0 0.2 0.4 0.6 0.8

Yield[arb]

12−10

10−10

8−10

6−10

4−10

2−10

1

210p2 lineshape

Pℓ(p1)

product

- 1/2→ - 1→

-(b) 5/2

Figure 8.10: R-matrix simulations for the 2p decay of 17Ne second-excited state through 16F(a) ground-state and (b) first-excited state. The line shape for the second proton p2 (black),the barrier penetration factor for the first proton p1, and their product (blue) are displayedfor both decay paths.

140

Chapter 9

Summary

I have used the invariant-mass method to study the proton-decaying nuclei labeled in Fig.

9.1. In this method, the decay energy and intrinsic width can be determined with high

precision for nuclear levels within a few MeV of their decay threshold and have lifetimes

shorter than 10−20 s (Γ ≥ 10 keV). For longer-lived nuclei, if the lifetime is not longer than

a nanoseconds, the decay energy can still be determined, however only an upper limit on

the intrinsic width can be measured. By studying these unbound, light nuclei we can better

understand their role in the production of elements in our universe.

A summary of the information measured in this work can be found in Table 9.1. As all

decay products are measured, this method is ideal for distinguishing the decay mode of a

given nuclear level. Both 8BIAS and 16Negs were found to decay by direct 2p emission. The

8BIAS was the first case proven to directly decay by 2p emission to the IAS in the daughter

because all other energy-allowed decay modes are forbidden by isospin. In 16Ne, the ground

state decayed by 2p to the ground state of 14O, but the first-excited state was found to

decay by both direct and sequential 2p, displaying interference effects between the two decay

paths. The IAS in 16F was believed to 2p decay to the IAS in 14N, analogous to the decay

of 8BIAS, however it appears that the larger Coulomb barrier in 16F highly suppresses this

decay mode. The dominant decay mode for 16FIAS remains unknown. For the A = 7 isobar,

141

C9

B8

Ne16 Ne17

B7

Be7

Li7

F16

Na17

Stable

Beta

2p or 2n

1p or 1n

virtual

N=46

810

16

Z=4

6

8

10

12

Z

N

Neutron Drip Line

Proton Drip Line

Figure 9.1: Chart of nuclides. Proton number increases towards the top of the figure, andneutron number increases towards the right. Stable nuclei are shown as black squares, andthe radioactive nuclei are shown as colored squares, indicating their dominate mode of decay.Nuclei studied in this work are labeled.

142

new measurements of the masses of three of the four nuclei by the invariant-mass method

has nearly eliminated the need for a cubic term in the Isobaric Mass Multiplet Equation. To

provide further information on this isobar, new experiments would require more than 1 order

of magnitude better resolution to provide constraints tight enough to address the need for

a small cubic term as predicted in ab initio calculations. We have made the first report on

the decay of 17Na, a collection of nucleons 3 neutrons deficient of the lightest, particle-bound

sodium isotope.

While this work was focused entirely on nuclei that decay by charged-particle emission,

this method has also be used to studying 1- and 2-neutron decay. The field now turns its

attention to the Facility for Rare Isotope Beams (FRIB), which will provide more exotic

and higher intensity radioactive ion beams than ever before produced on earth. While in

this new era experimental advances will be required to improve resolution and efficiency,

the invariant-mass method will continue to be a major tool in the study of the most exotic

nuclei. Progress will continue to be made with the invariant-mass method on both sides of

the nuclear chart.

143

Table 9.1: New spectroscopic information for nuclear levels discussed in this work. Excitationenergies, intrinsic widths and decay modes are provided when measured. For nuclear levelswith multiple decay modes, branching ratios are given as intensities out of 100 relative to thedominant decay mode. For ground-state decays, decay energy is given in place of excitationenergy.

Nucleus Excitation Energy Width Decay Mode(MeV) (keV)

7LiIAS 11.267 (15) 191 (41) 7LiIAS → p+6He, I = 1007LiIAS → n+(6LiIAS) → n+(6Lig.s.+γ), I = 88.7

7BeIAS 10.989 (15) 376 (47) 7BeIAS → p+(6LiIAS) → p+(6Lig.s.+γ), I = 1007BeIAS → n+6Be, I = 9.9

7Bg.s. 0.0 [Er = 2.036 (26)] 599 (113) 7Bg.s. → p+(6Beg.s.) → p+(2p+α)8B1+ 0.769 (25) 8B1+ → p+7Beg.s., I = 100

8B1+ → p+7Be1/2− , I = 1.18B3+ 2.320 (20) 8B1+ → p+7Beg.s.8BIAS 10.614 (20) <60 8BIAS → 2p+(6LiIAS) → 2p+(6Lig.s.+γ), I = 100

8BIAS → 2p+(6Li3+) → 2p+(d+α), I = 108BIAS → 2p+6Lig.s., I = 118BIAS → p+7Beg.s., I ≤ 7.5

9C1/2− 2.218 (11) 52 (11) 9C1/2− → p+8B9C5/2− 3.549 (20) 673 (50) 9C5/2− → p+8B

9C 4.40 (2) 9C → p+(8B1+) → p+(p+7Be)9C 5.69 (2) 9C → p+(8B3+) → p+(p+7Be)16F 7.67 (2) 16F → 2p+14Ng.s.

16F 10.26 (2) 16F → 2p+14Ng.s.

16Neg.s. 0.0 [Er = 1.466 (20)] <80 16Neg.s. → 2p+14Og.s.16Ne2+ 1.69 (2) 150 (50) 16Ne2+ → 2p+14Og.s.

16Ne2+ → p+(15Fg.s.) → p+(p+14Og.s.)16Ne2+ 7.60 (4) Seen in the 2p+14O channel, decay type unknown16Ne 8.37 (10) 320 (100) Seen in the 3p+13N channel, likely through 14O2+

16Ne 10.76 (20) 510 (230) Seen in the 3p+13N channel, likely through 14O2+

17Ne5/2− 1.764 (12) 17Ne5/2− → p+(16Fg.s.) → p+(p+15Og.s.)17Ne5/2+ 2.651 (12) Seen in the 2p+15O channel, decay type unknown17Ne9/2− 3.548 (20) Seen in the 2p+15O channel, decay type unknown

17Na 0.0 [Er ≤ 4.85 (6)] Seen in the 3p+14O, decay type unknown

144

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